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 (No. 333-228794) on Form S-8 of Aptose Biosciences Inc. (the “Company”) of our report dated March 12, 2019, with respect to the consolidated financial statements of Aptose Biosciences Inc. (the “Company”), which comprise the consolidated statements of financial position as at December 31, 2018 and December 31, 2017, the related consolidated statements of comprehensive loss and comprehensive loss, changes in shareholders’ equity and cash flows for each of the years in the two-year period ended December 31, 2018, and the related notes (collectively, the “consolidated financial statements”).

Our report dated March 12, 2019, contains an explanatory paragraph that states, without qualifying our opinion, that we draw attention to Note 2(a) and 15 to the consolidated financial statements, which indicates that the Company has retrospectively adopted United States generally accepted accounting principles (U.S. GAAP).

Chartered Professional Accountants, Licensed Public Accountants

March 12, 2019

Vaughan, Canada