EX-99.1

Exhibit 99.1

Cover Letter to Offer to Purchase and Letter of Transmittal

IF YOU DO NOT WANT TO TENDER YOUR LIMITED PARTNERSHIP INTERESTS FOR REPURCHASE AT

THIS TIME FOR ANY REASON, INCLUDING IF YOU ARE SUBJECT TO AN AGREEMENT NOT

TO TENDER YOUR INTERESTS FOR REPURCHASE DURING AN INITIAL HOLDING PERIOD, PLEASE DISREGARD

THIS NOTICE. THIS IS SIMPLY NOTIFICATION OF THE FUND’S TENDER OFFER.

January 23, 2013

Dear Investor of The Endowment Institutional TEI Fund W, L.P.:

We are writing to inform you of important dates related to the tender offer by The Endowment Institutional TEI Fund W, L. P. (the “Fund”). If you are not interested in selling your limited partnership interests at this time, please disregard this notice and take no action.

Please note that the sale of your limited partnership interests may be subject to a 2% Early Repurchase Fee if you decide to sell your interests in this offer.

The tender offer period will begin on January 23, 2013 and end on February 21, 2013. The purpose of the tender offer is to provide liquidity to investors who hold limited partnership interests. Limited partnership interests can be repurchased by means of a tender offer only during one of the Fund’s announced tender offers.

Should you wish to sell any of your limited partnership interests during this tender offer period, please complete the enclosed Letter of Transmittal (the two page Tender Offer Form will suffice) and contact your financial advisor who services your account for specific return instructions. Allow for additional processing time by your financial intermediary if necessary as the form must ultimately be received by the Fund’s Investment Adviser, Endowment Advisers, L.P. either by mail or by fax in good order by midnight, Central Standard Time, on February 21, 2013. If you do not wish to tender limited partnership interests for repurchase for any reason, including if you are subject to an agreement not to tender your interests for repurchase during an initial holding period, simply disregard this notice. No Action is Required if You Do Not Wish to Tender Your Interests for Repurchase at This Time.

If you have any questions, please refer to the attached Offer to Purchase document, which contains additional important information about the tender offer, or call the Support Desk of the Investment Adviser at (800) 725-9456.

Sincerely,

The Endowment Institutional TEI Fund W, L.P.

Cover Letter to Offer to Purchase and Letter of Transmittal

Clients of U.S. Trust Portfolio Managers

IF YOU DO NOT WANT TO TENDER YOUR LIMITED PARTNERSHIP INTERESTS FOR REPURCHASE AT

THIS TIME FOR ANY REASON, INCLUDING IF YOU ARE SUBJECT TO AN AGREEMENT NOT

TO TENDER YOUR INTERESTS FOR REPURCHASE DURING AN INITIAL HOLDING PERIOD, PLEASE DISREGARD

THIS NOTICE. THIS IS SIMPLY NOTIFICATION OF THE FUND’S TENDER OFFER.

January 23, 2013

Dear Investor of The Endowment Institutional TEI Fund W, L.P.:

We are writing to inform you of important dates related to the tender offer by The Endowment Institutional TEI Fund W, L.P. (the “Fund”). If you are not interested in selling your limited partnership interests at this time, please disregard this notice and take no action.

Please note that the sale of your limited partnership interests may be subject to a 2% Early Redemption Fee if you decide to sell your interests in this offer.

The tender offer period will begin on January 23, 2013 and end on February 21, 2013. The purpose of the tender offer is to provide liquidity to investors who hold limited partnership interests. Limited partnership interests can be repurchased by means of a tender offer only during one of the Fund’s announced tender offers.

Should you wish to sell any of your limited partnership interests during this tender offer period, please contact your Portfolio Manager who will provide you a customized Tender Offer Form for your account. The Letter of Transmittal contained in this Offer to Purchase includes a sample Tender Offer Form for your reference only. The Tender Offer Form generated for your account will need to be signed and returned to your Portfolio Manager. Upon receiving the signed form, your Portfolio Manager will then submit the order. Your Portfolio Manager must submit the form by midnight, Central Standard Time, on February 21, 2013. If you do not wish to tender limited partnership interests for repurchase for any reason, including if you are subject to an agreement not to tender your interests for repurchase during an initial holding period, simply disregard this notice. No Action is Required if You Do Not Wish to Tender Your Interests for Repurchase at This Time.

If you have any questions, please refer to the attached Offer to Purchase document, which contains additional important information about the tender offer, or call your Portfolio Manager.

Sincerely,

The Endowment Institutional TEI Fund W, L.P.