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

    <schemaVersion>X0306</schemaVersion>

    <documentType>4</documentType>

    <periodOfReport>2017-05-19</periodOfReport>

    <notSubjectToSection16>0</notSubjectToSection16>

    <issuer>
        <issuerCik>0001674760</issuerCik>
        <issuerName>Goldman Sachs Private Middle Market Credit LLC</issuerName>
        <issuerTradingSymbol>NONE</issuerTradingSymbol>
    </issuer>

    <reportingOwner>
        <reportingOwnerId>
            <rptOwnerCik>0001635552</rptOwnerCik>
            <rptOwnerName>McGovern, Brendan</rptOwnerName>
        </reportingOwnerId>
        <reportingOwnerAddress>
            <rptOwnerStreet1>200 WEST STREET</rptOwnerStreet1>
            <rptOwnerStreet2></rptOwnerStreet2>
            <rptOwnerCity>NEW YORK</rptOwnerCity>
            <rptOwnerState>NY</rptOwnerState>
            <rptOwnerZipCode>10282</rptOwnerZipCode>
            <rptOwnerStateDescription></rptOwnerStateDescription>
        </reportingOwnerAddress>
        <reportingOwnerRelationship>
            <isDirector>0</isDirector>
            <isOfficer>1</isOfficer>
            <isTenPercentOwner>0</isTenPercentOwner>
            <isOther>0</isOther>
            <officerTitle>See Remarks</officerTitle>
            <otherText></otherText>
        </reportingOwnerRelationship>
    </reportingOwner>

    <nonDerivativeTable>
        <nonDerivativeTransaction>
            <securityTitle>
                <value>Common Units of limited liability company interests</value>
            </securityTitle>
            <transactionDate>
                <value>2017-05-19</value>
            </transactionDate>
            <deemedExecutionDate></deemedExecutionDate>
            <transactionCoding>
                <transactionFormType>4</transactionFormType>
                <transactionCode>P</transactionCode>
                <equitySwapInvolved>0</equitySwapInvolved>
            </transactionCoding>
            <transactionAmounts>
                <transactionShares>
                    <value>40.47</value>
                </transactionShares>
                <transactionPricePerShare>
                    <value>98.8336</value>
                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
                    <value>A</value>
                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>254.28</value>
                    <footnoteId id="F1"/>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </nonDerivativeTransaction>
    </nonDerivativeTable>

    <derivativeTable></derivativeTable>

    <footnotes>
        <footnote id="F1">On the date hereof, Mr. McGovern owns 213.81 Common Units of limited liability company interests (the &quot;Common Units&quot;) of Goldman Sachs Private Middle Market Credit LLC (the &quot;Company&quot;), after taking into account the final allocation of units as part of a prior capital call, which number was estimated and reported on a prior filing. In addition, on May 19, 2017, the Company issued a capital call notice to its investors that have committed to purchase Common Units. Pursuant to such capital call notice, Mr. McGovern is obligated to make a capital contribution of $4,000 on May 26, 2017 out of his $100,000 aggregate capital commitment to the Company, and the Company is obligated to issue approximately 40.47 Common Units to Mr. McGovern, provided such capital contribution is funded timely.</footnote>
    </footnotes>

    <remarks>Chief Executive Officer and President</remarks>

    <ownerSignature>
        <signatureName>/s/ Neena A. Reddy, Attorney-in-fact</signatureName>
        <signatureDate>2017-05-23</signatureDate>
    </ownerSignature>
</ownershipDocument>
