<?xml version="1.0"?>
<ownershipDocument>

    <schemaVersion>X0306</schemaVersion>

    <documentType>4</documentType>

    <periodOfReport>2022-09-19</periodOfReport>

    <notSubjectToSection16>0</notSubjectToSection16>

    <issuer>
        <issuerCik>0000882361</issuerCik>
        <issuerName>Aptose Biosciences Inc.</issuerName>
        <issuerTradingSymbol>APTO</issuerTradingSymbol>
    </issuer>

    <reportingOwner>
        <reportingOwnerId>
            <rptOwnerCik>0001761487</rptOwnerCik>
            <rptOwnerName>Rice William G.</rptOwnerName>
        </reportingOwnerId>
        <reportingOwnerAddress>
            <rptOwnerStreet1>C/O APTOSE BIOSCIENCES INC.</rptOwnerStreet1>
            <rptOwnerStreet2>251 CONSUMERS ROAD, SUITE 1105</rptOwnerStreet2>
            <rptOwnerCity>TORONTO</rptOwnerCity>
            <rptOwnerState>A6</rptOwnerState>
            <rptOwnerZipCode>M2J 4R3</rptOwnerZipCode>
            <rptOwnerStateDescription>ONTARIO, CANADA</rptOwnerStateDescription>
        </reportingOwnerAddress>
        <reportingOwnerRelationship>
            <isDirector>1</isDirector>
            <isOfficer>1</isOfficer>
            <isTenPercentOwner>0</isTenPercentOwner>
            <isOther>0</isOther>
            <officerTitle>Chair, President &amp; CEO</officerTitle>
            <otherText></otherText>
        </reportingOwnerRelationship>
    </reportingOwner>

    <nonDerivativeTable>
        <nonDerivativeTransaction>
            <securityTitle>
                <value>Common Shares</value>
            </securityTitle>
            <transactionDate>
                <value>2022-09-19</value>
            </transactionDate>
            <deemedExecutionDate></deemedExecutionDate>
            <transactionCoding>
                <transactionFormType>4</transactionFormType>
                <transactionCode>P</transactionCode>
                <equitySwapInvolved>0</equitySwapInvolved>
            </transactionCoding>
            <transactionAmounts>
                <transactionShares>
                    <value>10000</value>
                </transactionShares>
                <transactionPricePerShare>
                    <value>0.6834</value>
                    <footnoteId id="F1"/>
                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
                    <value>A</value>
                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>363252</value>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </nonDerivativeTransaction>
    </nonDerivativeTable>

    <derivativeTable></derivativeTable>

    <footnotes>
        <footnote id="F1">The purchase price reported in Column 4 represents a weighted average price.  These shares were purchased in multiple transactions at prices ranging from  $0.6834 to $0.6835, inclusive.  The reporting person undertakes to provide to Aptose Biosciences Inc., any security holder of Aptose Biosciences Inc., or the staff of the Securities and Exchange Commission, upon request, full information regarding the number of shares purchased at each separate price within the ranges set forth above.</footnote>
    </footnotes>

    <remarks>Exhibit 24.2  Power of Attorney</remarks>

    <ownerSignature>
        <signatureName>/s/ Leslie Auld as attorney-in-fact for William G. Rice</signatureName>
        <signatureDate>2022-09-20</signatureDate>
    </ownerSignature>
</ownershipDocument>
