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

    <schemaVersion>X0306</schemaVersion>

    <documentType>4</documentType>

    <periodOfReport>2013-03-27</periodOfReport>

    <notSubjectToSection16>0</notSubjectToSection16>

    <issuer>
        <issuerCik>0001030749</issuerCik>
        <issuerName>GULFMARK OFFSHORE INC</issuerName>
        <issuerTradingSymbol>GLF</issuerTradingSymbol>
    </issuer>

    <reportingOwner>
        <reportingOwnerId>
            <rptOwnerCik>0001206113</rptOwnerCik>
            <rptOwnerName>STREETER BRUCE A</rptOwnerName>
        </reportingOwnerId>
        <reportingOwnerAddress>
            <rptOwnerStreet1>10111 RICHMOND AVE, STE 340</rptOwnerStreet1>
            <rptOwnerStreet2></rptOwnerStreet2>
            <rptOwnerCity>HOUSTON</rptOwnerCity>
            <rptOwnerState>TX</rptOwnerState>
            <rptOwnerZipCode>77042</rptOwnerZipCode>
            <rptOwnerStateDescription></rptOwnerStateDescription>
        </reportingOwnerAddress>
        <reportingOwnerRelationship>
            <isDirector>1</isDirector>
            <isOfficer>1</isOfficer>
            <isTenPercentOwner>0</isTenPercentOwner>
            <isOther>0</isOther>
            <officerTitle>President and CEO</officerTitle>
            <otherText></otherText>
        </reportingOwnerRelationship>
    </reportingOwner>

    <nonDerivativeTable>
        <nonDerivativeTransaction>
            <securityTitle>
                <value>Common Stock</value>
                <footnoteId id="F1"/>
            </securityTitle>
            <transactionDate>
                <value>2013-03-27</value>
            </transactionDate>
            <deemedExecutionDate></deemedExecutionDate>
            <transactionCoding>
                <transactionFormType>4</transactionFormType>
                <transactionCode>F</transactionCode>
                <equitySwapInvolved>0</equitySwapInvolved>
            </transactionCoding>
            <transactionAmounts>
                <transactionShares>
                    <value>8701</value>
                </transactionShares>
                <transactionPricePerShare>
                    <value>39.183</value>
                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
                    <value>D</value>
                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>514097</value>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </nonDerivativeTransaction>
    </nonDerivativeTable>

    <derivativeTable></derivativeTable>

    <footnotes>
        <footnote id="F1">Under the terms of the GulfMark 1997 Incentive Equity Plan and 2010 Omnibus Equity Incentive Plan, shares were sold to the Company to satisfy the tax obligation on restricted shares which restrictions lapsed.</footnote>
    </footnotes>

    <remarks></remarks>

    <ownerSignature>
        <signatureName>/s/ Bruce Streeter</signatureName>
        <signatureDate>2013-03-29</signatureDate>
    </ownerSignature>
</ownershipDocument>
