<?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>0001650476</rptOwnerCik>
            <rptOwnerName>Payne Fletcher</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>0</isDirector>
            <isOfficer>1</isOfficer>
            <isTenPercentOwner>0</isTenPercentOwner>
            <isOther>0</isOther>
            <officerTitle>SR VP &amp; Chief Fin. Officer</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>0.6775</value>
                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
                    <value>A</value>
                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>10000</value>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>I</value>
                </directOrIndirectOwnership>
                <natureOfOwnership>
                    <value>Fletcher Payne 2019 Trust</value>
                </natureOfOwnership>
            </ownershipNature>
        </nonDerivativeTransaction>
    </nonDerivativeTable>

    <derivativeTable></derivativeTable>

    <footnotes></footnotes>

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

    <ownerSignature>
        <signatureName>/s/ Leslie Auld as attorney-in-fact for Fletcher Payne</signatureName>
        <signatureDate>2022-09-21</signatureDate>
    </ownerSignature>
</ownershipDocument>
