Exhibit 23.1

KPMG LLP

Vaughan Metropolitan Centre 100 New Park Place, Suite 1400 Vaughan, ON L4K 0J3

Canada

Tel 905-265-5900

Fax 905-265-6390

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 (No. 333-235730) on Form S-3 of common shares or warrants or any combination of those securities, either individually or in units, up to an aggregate initial offering price of $200,000,000 and by reference in the registration statements on Form S-8 (No. 333- 228794) and (No.333-205158) related to the stock incentive plan of Aptose Biosciences Inc. (the Company), of our report dated March 10, 2020, on the consolidated financial statements of the Company, which comprise the consolidated statements of financial position as at December 31, 2019 and December 31, 2018, 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, 2019, and the related notes, (collectively, the consolidated financial statements), and to the reference to our firm under the heading “Experts” in the registration statement.

Our report dated March 10, 2020, on the consolidated financial statements referred to above contains an explanatory paragraph indicating the Company has changed its method of accounting for leases on January 1, 2019, due to the adoption of Financial Accounting Standards Board, ASU No. 2016-02, Leases (Topic 842).

Chartered Professional Accountants, Licensed Public Accountants

March 10, 2020

Vaughan, Canada

KPMG LLP is a Canadian limited liability partnership and a member firm of the KPMG network of independent member firms affiliated with KPMG International Cooperative (“KPMG International”), a Swiss entity. KPMG Canada provides services to KPMG LLP.