LIMITED POWER OF ATTORNEY
                  FOR SECTION 16 REPORTING OBLIGATIONS

        Know all by these presents, that the undersigned hereby makes,
constitutes and appoints Manuel C. Alves Aivado, Rachael Perry and Susan L.
Drexler, signing singly and each acting individually, as the undersigned's true
and lawful attorney-in-fact with full power and authority as hereinafter
described to:

        (1)    execute for and on behalf of the undersigned, in the
undersigned's capacity as an officer of Aileron Therapeutics, Inc. (the
"Company"), Forms 3, 4, and 5 (including any amendments thereto) in accordance
with Section 16(a) of the Securities Exchange Act of 1934 and the rules
thereunder (the "Exchange Act");

        (2)    do and perform any and all acts for and on behalf of the
undersigned which may be necessary or desirable to prepare, complete and
execute any such Form 3, 4, or 5, prepare, complete and execute any amendment
or amendments thereto, and timely deliver and file such form with the United
States Securities and Exchange Commission (the "SEC") and any stock exchange or
similar authority, including without limitation the filing of a Form ID, Update
Passphrase, or any other application materials to enable the undersigned to
gain or maintain access to the Electronic Data Gathering, Analysis and
Retrieval system of the SEC;

        (3)    seek or obtain, as the undersigned's representative and on the
undersigned's behalf, information regarding transactions in the Company's
securities from any third party, including brokers, employee benefit plan
administrators and trustees, and the undersigned hereby authorizes any such
person to release any such information to such attorney-in-fact and approves
and ratifies any such release of information; and

        (4)    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, are not assuming nor relieving, nor is the Company assuming nor
relieving, any of the undersigned's responsibilities to comply with Section 16
of the Exchange Act. The undersigned acknowledges that neither the Company nor
the foregoing attorneys-in-fact assume (i) any liability for the undersigned's
responsibility to comply with the requirement of the Exchange Act, (ii) any
liability of the undersigned for any failure to comply with such requirements,
or (iii) any obligation or liability of the undersigned for profit disgorgement
under Section 16(b) 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.

        IN WITNESS WHEREOF, the undersigned has caused this Power of Attorney
to be executed as of June 15, 2022.

                                   /s/ Jeffrey Bailey
                                   ----------------------------
                                   Name: Jeffrey Bailey