Exhibit 32.1

Certification Pursuant to 18 U.S.C. Sec. 1350

In connection with the filing by Tremont Mortgage Trust (the “Company”) of the Quarterly Report on Form 10-Q for the period ended September 30, 2017 (the “Report”), each of the undersigned hereby certifies, to the best of his knowledge:

1) The Report fully complies with the requirements of Section 13(a) or 15(d) of the Securities Exchange Act of 1934; and

2) The information contained in the Report fairly presents, in all material respects, the financial condition and results of operations of the Company.

/s/ Adam D. Portnoy	/s/ David M. Blackman
Adam D. Portnoy Managing Trustee	David M. Blackman Chief Executive Officer

/s/ Barry M. Portnoy	/s/ G. Douglas Lanois
Barry M. Portnoy Managing Trustee	G. Douglas Lanois Chief Financial Officer and Treasurer

Date:    November 14, 2017