EXHIBIT E

FORM OF LETTER FROM THE FUND TO PARTNERS IN CONNECTION WITH NOTICE AND
ACCEPTANCE OF OFFERS OF TENDER

[______ __, 20__]

Dear Partner:

Hatteras Core Alternatives TEI Institutional Fund, L.P. (the “Fund”) has received and accepted your tender request for all or some of your limited partnership units in the Fund (your “Units”).

In accordance with the terms of the tender offer, and the Fund’s agreement to repurchase all or a part of your Units, you have been issued a promissory note (the “Note”) entitling you to receive an initial amount of at least 95% (100% if you did not tender your entire position) of the value of the repurchased Units. The value will be based on the unaudited net asset value of the Fund as of June 30, 2020, in accordance with the terms of the tender offer.

The Note will be held by UMB Fund Services, Inc. (“UMBFS”) on your behalf. Upon written request by you to UMBFS, the Note will be mailed to your address of record. The amount repurchased is expected to be paid in cash by wire transfer between August and September 2020 to the account designated in your Letter of Transmittal after final valuations are calculated on or about July 20, 2020.

If you tendered your entire position, the terms of the Note and the tender offer state that a post-audit payment representing the balance of the repurchase amount, if any, will be paid to you promptly after the completion of the Fund’s next annual audit. We expect that the annual audit of the Fund’s financial statements will be completed by the end of May 2021.

If you tendered only some of your Units, you remain an investor in the Fund with respect to your Units that were not repurchased by the Fund.

If you wish to rescind your tender request, you may do so by June 30, 2020. Please contact your Financial Advisor for more information. The next tender offer period is expected to commence on or about June 25, 2020.

Should you have any questions, please feel free to contact your Financial Advisor or the Fund's Tender Offer Administrator, UMB Fund Services, at (800) 504-9070.

Sincerely,

Hatteras Core Alternatives TEI Institutional Fund, L.P.

 FORM OF LETTER FROM THE FUND TO PARTNERS IN CONNECTION WITH NOTICE AND
ACCEPTANCE OF PRO RATA (OVERSUBSCRIBED) OFFERS OF TENDER

[______ __, 20__]

Dear Partner:

Hatteras Core Alternatives TEI Institutional Fund, L.P. (the "Fund") has received and accepted approximately [__]% of your tender request for your limited partnership units in the Fund (your “Units”). The exact percentage will not be available until final valuations are calculated on or about July 20, 2020.

For the tender period ended, April 24, 2020, approximately [__]% of Units in the Fund were submitted for tender and the offer was oversubscribed. In accordance with the terms of the tender offer, you have been issued a promissory note (the “Note”) entitling you to receive your pro-rata share of your tender request. Thus, the Fund will repurchase from you approximately [__]% of your requested redemption amount. For example, if you requested $100,000 in redemption, you will receive approximately $[_____].

The Note will be held by UMB Fund Services, Inc. (“UMBFS”) on your behalf. No action is required by you at this time. However, if you would like to receive a copy of the Note, please submit a written request to UMBFS and a copy will be mailed to your address of record. The amount repurchased is expected to be paid in cash by wire transfer between August and September 2020 to the account designated in your Letter of Transmittal. Please note that because a portion of your capital remains with the Fund, you are still an investor in the Fund and a 5% holdback will not be applied.

If you wish to rescind your tender request, you may do so by June 30, 2020. Please contact your Financial Advisor for more information. The next tender offer period is expected to commence on or about June 25, 2020.

Should you have any questions, please feel free to contact your Financial Advisor or the Fund's Tender Offer Administrator, UMB Fund Services, at (800) 504-9070.

Sincerely,

Hatteras Core Alternatives TEI Institutional Fund, L.P.