
<PAGE>   1
                                                                EXHIBIT 99(a)(4)

 
                           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.
 
                                                               November 25, 1997
To Brokers, Dealers, Commercial Banks,
  Trust Companies and Other Nominees:
 
     We have been appointed by DSL Acquisition Corporation (the "Purchaser"), a
Delaware corporation and a direct wholly owned subsidiary of Texas Instruments
Incorporated, a Delaware corporation ("Parent"), to act as Dealer Manager in
connection with its offer to purchase all 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 to Purchase dated November 25, 1997 (the "Offer to Purchase") and
in the related Letter of Transmittal (which, together with any amendments and
supplements, collectively constitute the "Offer"), copies of which are enclosed
herewith.
 
     For your information and for forwarding to your clients for whose accounts
you hold Shares registered in your name or in the name of your nominee, we are
enclosing the following documents:
 
          1. Offer to Purchase;
 
          2. Letter of Transmittal for your use and for the information of your
     clients, together with Guidelines for Certification of Taxpayer
     Identification Number on Substitute Form W-9 providing information relating
     to backup federal income tax withholding;
 
          3. Notice of Guaranteed Delivery to be used to accept the Offer if the
     Shares and all other required documents cannot be delivered to the
     Depositary by the Expiration Date (as defined in the Offer to Purchase);
 
          4. A form of letter which may be sent to your clients for whose
     accounts you hold Shares registered in your name or in the name of your
     nominee, with space provided for obtaining such clients' instructions with
     regard to the Offer;
 
          5. Solicitation/Recommendation Statement on Schedule 14D-9 issued by
     the Company; and
 
          6. Return envelope addressed to ChaseMellon Shareholder Services,
     L.L.C., as the Depositary.
 
     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
<PAGE>   2
 
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 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.
 
     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.
 
     Upon the terms and subject to the conditions of the Offer (including, if
the Offer is extended or amended, the terms and conditions of any such extension
or amendment), the Purchaser will purchase, by accepting for payment, and will
pay for, all Shares validly tendered and not properly withdrawn prior to the
Expiration Date (as defined in the Offer to Purchase) if, as and when the
Purchaser gives oral or written notice to the Depositary of the Purchaser's
acceptance of the tenders of such Shares for payment pursuant to the Offer.
Payment for Shares purchased pursuant to the Offer will be made only after
timely receipt by the Depositary of (i) certificates evidencing such Shares or
timely confirmation of a book-entry transfer of such Shares into the
Depositary's account at one of the Book-Entry Transfer Facilities (as defined in
the Offer to Purchase), (ii) a properly completed and duly executed Letter of
Transmittal (or facsimile thereof) or, in the case of a book-entry transfer, an
Agent's Message (as defined in the Offer to Purchase) and (iii) any other
documents required by 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.
 
     If a stockholder desires to tender Shares pursuant to the Offer and such
stockholder's stock certificates are not immediately available or time will not
permit all required documents to reach the Depositary prior to the Expiration
Date, or the procedures for book-entry transfer cannot be completed on a timely
basis, such Shares may nevertheless be tendered by following the guaranteed
delivery procedures specified in Section 3 of the Offer to Purchase.
 
     Neither Parent nor the Purchaser will pay any fees or commissions to any
broker, dealer or other person (other than the Dealer Manager, the Information
Agent and the Depositary as described in the Offer to Purchase) in connection
with the solicitation of tenders of Shares pursuant to the Offer. The Purchaser
will, however, upon request, reimburse brokers, dealers, commercial banks and
trust companies for reasonable expenses incurred by them in forwarding materials
to their customers. The Purchaser will pay all stock transfer taxes applicable
to its purchase of Shares pursuant to the Offer, subject to Instruction 6 of the
Letter of Transmittal.
 
     YOUR PROMPT ACTION IS REQUESTED. WE URGE YOU TO CONTACT YOUR CLIENTS AS
PROMPTLY AS POSSIBLE. 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.
<PAGE>   3
 
     Any inquiries you may have with respect to the Offer may be addressed to
the Information Agent or the undersigned at the addresses and telephone numbers
set forth on the back cover page of the Offer to Purchase. Requests for
additional copies of the enclosed materials may be directed to the Information
Agent or the Dealer Managers.
 
                                        Very truly yours,
 
                                        MORGAN STANLEY & CO. INCORPORATED
 
     NOTHING CONTAINED HEREIN OR IN THE ENCLOSED DOCUMENTS SHALL RENDER YOU OR
ANY PERSON THE AGENT OF THE PURCHASER, PARENT, THE COMPANY, ANY AFFILIATE OF THE
COMPANY, THE DEALER MANAGER, THE INFORMATION AGENT OR THE DEPOSITARY, OR
AUTHORIZE YOU OR ANY OTHER PERSON TO USE ANY DOCUMENT OR MAKE ANY STATEMENT ON
BEHALF OF ANY OF THEM IN CONNECTION WITH THE OFFER OTHER THAN THE DOCUMENTS
ENCLOSED HEREWITH AND THE STATEMENTS CONTAINED THEREIN.
