
<PAGE>   1
                                                               EXHIBIT 99(a)(5)

 
                           Offer to Purchase for Cash
                     All Outstanding Shares of Common Stock
                                       of
 
                        Amati Communications Corporation
                                       at
                              $20.00 Net Per Share
                                       by
 
                          DSL Acquisition Corporation
                        a direct wholly owned subsidiary
                                       of
 
                         Texas Instruments Incorporated
 
  THE OFFER AND WITHDRAWAL RIGHTS WILL EXPIRE AT 12:00 MIDNIGHT, NEW YORK CITY
       TIME, ON TUESDAY, DECEMBER 23, 1997, UNLESS THE OFFER IS EXTENDED.
 
To Our Clients:
 
     Enclosed for your consideration are the Offer to Purchase dated November
25, 1997 (the "Offer to Purchase") and the related Letter of Transmittal (which,
together with any amendments and supplements, collectively constitute the
"Offer") and other materials relating to the Offer by DSL Acquisition
Corporation (the "Purchaser"), a Delaware corporation and a direct wholly owned
subsidiary of Texas Instruments Incorporated, a Delaware corporation ("Parent"),
to purchase all of the outstanding shares of common stock, par value $.20 per
share (the "Shares"), of Amati Communications Corporation, a Delaware
corporation (the "Company"), at $20.00 per Share, net to the seller in cash,
without interest, upon the terms and subject to the conditions set forth in the
Offer. Also enclosed is the letter to stockholders of the Company from the
Chairman of the Board and the President and Chief Executive Officer of the
Company accompanied by the Company's Solicitation/Recommendation Statement on
Schedule 14D-9. This material is being sent to you as the beneficial owner of
Shares held by us for your account but not registered in your name. A tender of
such Shares can be made only by us as the holder of record and pursuant to your
instructions. The Letter of Transmittal accompanying this letter is furnished to
you for your information only and cannot be used by you to tender Shares held by
us for your account.
 
     We request instructions as to whether you wish to have us tender any or all
of the Shares held by us for your account, upon the terms and subject to the
conditions set forth in the Offer.
 
     Your attention is directed to the following:
 
          1. The tender price is $20.00 per Share, net to the seller in cash,
     without interest, upon the terms and subject to the conditions of the
     Offer.
 
          2. The Offer and withdrawal rights will expire at 12:00 midnight, New
     York City time, on Tuesday, December 23, 1997, unless the Offer is
     extended.
 
          3. The Offer is being made pursuant to an Agreement and Plan of
     Merger, dated as of November 19, 1997 (the "Merger Agreement"), by and
     among the Company, Parent and the Purchaser. The Merger Agreement provides
     that, among other things, following the consummation of the Offer and the
     satisfaction or waiver of the other conditions set forth in the Merger
     Agreement, the Purchaser will be merged with and into the Company (the
     "Merger"). Following the Merger, the Company will continue as the surviving
     corporation and will be a direct wholly owned subsidiary of Parent. At the
     effective time of the Merger, each outstanding Share (other than Shares
     owned by Parent, the Purchaser or any other wholly owned subsidiary of
     Parent and Shares held by stockholders who have demanded and perfected
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     dissenter's rights under Delaware law) will be converted into the right to
     receive the per Share price paid in the Offer in cash, without interest.
 
          4. The Board of Directors of the Company has unanimously approved the
     Merger Agreement, the Offer and the Merger, and has determined that the
     Offer and the Merger are fair to and in the best interests of the Company's
     stockholders and unanimously recommends that stockholders accept the Offer
     and tender their Shares in the Offer.
 
          5. The Offer is conditioned upon, among other things, there being
     validly tendered and not withdrawn prior to the expiration of the Offer,
     that number of Shares which, together with any Shares beneficially owned by
     Parent or the Purchaser, represent at least a majority of the Shares
     outstanding on a fully diluted basis. Subject to the terms of the Merger
     Agreement, the Offer is also subject to other terms and conditions set
     forth in the Offer to Purchase. Pursuant to the Merger Agreement, the
     Purchaser has agreed to extend the Offer from time to time until February
     23, 1998 (as such date may be extended pursuant to the terms of the Merger
     Agreement), if at the initial Expiration Date (as defined in the Offer to
     Purchase), or any extension thereof, all conditions to the Offer have not
     been satisfied or waived. Any or all conditions to the Offer may be waived
     by the Purchaser.
 
          6. Any stock transfer taxes applicable to the sale of Shares to the
     Purchaser pursuant to the Offer will be paid by the Purchaser, except as
     otherwise provided in Instruction 6 of the Letter of Transmittal.
 
     In order to tender Shares pursuant to the Offer, a properly completed and
duly executed Letter of Transmittal (or facsimile thereof), with any required
signature guarantees, or an Agent's Message (in the case of any book-entry
transfer), and any other documents required by the Letter of Transmittal, should
be sent to the Depositary, and either certificates representing the tendered
Shares should be delivered or such Shares must be delivered to the Depositary
pursuant to the procedures for book-entry transfers, all in accordance with the
instructions set forth in the Letter of Transmittal and the Offer to Purchase.
 
     The Offer is being made to all holders of Shares. The Offer is not being
made to, nor will tenders be accepted from or on behalf of, holders of Shares in
any jurisdiction in which the making of the Offer or acceptance thereof would
not be in compliance with the laws of such jurisdiction. In any jurisdiction
where the securities, blue sky or other laws require the Offer to be made by a
licensed broker or dealer, the Offer shall be deemed to be made on behalf of the
Purchaser by Morgan Stanley & Co. Incorporated or one or more registered brokers
or dealers licensed under the laws of such jurisdictions.
 
     If you wish to have us tender any or all of the Shares held by us for your
account, please so instruct us by completing, executing and returning to us the
instruction form set forth below. Please forward your instructions to us in
ample time to permit us to submit a tender on your behalf prior to the
expiration of the Offer. If you authorize the tender of your Shares, all such
Shares will be tendered unless otherwise specified on the instruction form set
forth below.
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                          Instructions with Respect to
                           Offer to Purchase for Cash
                     All Outstanding Shares of Common Stock
                                       of
 
                        Amati Communications Corporation
                                       by
 
                          DSL Acquisition Corporation
                        a direct wholly owned subsidiary
                                       of
 
                         Texas Instruments Incorporated
 
     The undersigned acknowledge(s) receipt of your letter and the enclosed
Offer to Purchase dated November 25, 1997 and the related Letter of Transmittal,
in connection with the offer by DSL Acquisition Corporation, a Delaware
corporation and a direct wholly owned subsidiary of Texas Instruments
Incorporated, a Delaware corporation, to purchase for cash all outstanding
shares of common stock, par value $.20 per share (the "Shares"), of Amati
Communications Corporation, a Delaware corporation.
 
     This will instruct you to tender the number of Shares indicated below (or
if no number is indicated below, all Shares) that are held by you for the
account of the undersigned, upon the terms and subject to the conditions set
forth in the Offer and the related Letter of Transmittal.
 
Dated:  __________________ , 199 __
 
                        NUMBER OF SHARES TO BE TENDERED:
                            ________________ SHARES*
 
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                                  Signature(s)
 
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                              Please Print Name(s)
 
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                            Please Print Address(es)
 
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                       Area Code and Telephone Number(s)
 
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                Tax Identification or Social Security Number(s)
 
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* I (We) understand that if I (we) sign this instruction form without indicating
  a lesser number of Shares in the space above, all Shares held by you for my
  (our) account will be tendered.
