Power of Attorney

     Know all by these presents, that the undersigned hereby constitutes and
appoints each of Margaret Siegel and Stephen Turkowiak, signing singly, and with
full power of substitution, as the undersigned's true and lawful attorney-in-
fact to:

          (1) execute for and on behalf of the undersigned, in the undersigned's
          capacity as an officer and/or director of TCR2 Therapeutics Inc., a
          Delaware corporation (the "Company"), from time to time the following
          Securities and Exchange Commission (the "SEC") forms: Forms 3, 4, and
          5 in accordance with Section 16(a) of the Securities Exchange Act of
          1934, as amended (the "Exchange Act"), and the rules thereunder;

          (2) do and perform any and all acts for and on behalf of the
          undersigned which may be necessary or desirable to complete and
          execute any such Form 3, 4, or 5, complete and execute any amendment
          or amendments thereto, and timely file such form with the SEC and any
          stock exchange or similar authority; and

          (3) take any other action of any type whatsoever in connection with
          the foregoing which, in the opinion of such attorney-in-fact, may be
          of benefit to, in the best interest of, or legally required by, the
          undersigned, it being understood that the documents executed by such
          attorney-in-fact on behalf of the undersigned pursuant to this Power
          of Attorney shall be in such form and shall contain such terms and
          conditions as such attorney-in-fact may approve in such attorney-in-
          fact's discretion.

     The undersigned hereby grants to each such attorney-in-fact full power and
authority to do and perform any and every act and thing whatsoever requisite,
necessary, or proper to be done in the exercise of any of the rights and powers
herein granted, as fully to all intents and purposes as the undersigned might or
could do if personally present, with full power of substitution or revocation,
hereby ratifying and confirming all that such attorney-in-fact, or such
attorney-in-fact's substitute or substitutes, shall lawfully do or cause to be
done by virtue of this power of attorney and the rights and powers herein
granted. The undersigned acknowledges that the foregoing attorneys-in-fact, in
serving in such capacity at the request of the undersigned, is not assuming, nor
is the Company assuming, any of the undersigned's responsibilities to comply
with Section 16 of the Exchange Act.

     This Power of Attorney shall remain in full force and effect until the
undersigned is no longer required to file Forms 3, 4, and 5 with respect to the
undersigned's holdings of and transactions in securities issued by the Company,
unless earlier revoked by the undersigned in a signed writing delivered to the
foregoing attorneys-in- fact. This Power of Attorney shall expire as to any
individual attorney-in-fact if such attorney-in-fact ceases to be an officer of
the Company. This Power of Attorney may be filed with the SEC as a confirming
statement of the authority granted herein.

     IN WITNESS WHEREOF, the undersigned has caused this Power of Attorney to be
executed as of this 27th day of June, 2022.


              /s/ Eric Sullivan
              -------------------------------

Printed Name: Eric Sullivan