Exhibit 23.1

KPMG LLP

100 New Park Place, Suite 1400

Vaughan, ON L4K 0J3

Tel 905-265 5900

Fax 905-265 6390

www.kpmg.ca

Consent of Independent Registered Public Accounting Firm

The Board of Directors

Aptose Biosciences Inc.

We, KPMG LLP, consent to the incorporation by reference in the registration statement on Form S-3 (No. 333-235730) and registration statements on Form S-8 (No. 333-228794 and No. 333-205158) of Aptose Biosciences Inc. (the Company), of our report dated March 23, 2021, on the consolidated financial statements of the Company, which comprise the consolidated statements of financial position as at December 31, 2020 and December 31, 2019, the related consolidated statements of loss and comprehensive loss, changes in shareholders’ equity and cash flows for each of the years in the two-year period ended December 31, 2020, and the related notes, and to the reference to our firm under the heading “Experts” in the registration statements.

Chartered Professional Accountants, Licensed Public Accountants

March 23, 2021

Vaughan, Canada

KPMG LLP, an Ontario limited liability partnership and member firm of the KPMG global organization of independent member firms affiliated with KPMG International Limited, a private English company limited by guarantee. KPMG Canada provides services to KPMG LLP.