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

    <schemaVersion>X0306</schemaVersion>

    <documentType>4</documentType>

    <periodOfReport>2018-11-05</periodOfReport>

    <notSubjectToSection16>0</notSubjectToSection16>

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

    <reportingOwner>
        <reportingOwnerId>
            <rptOwnerCik>0001635536</rptOwnerCik>
            <rptOwnerName>Lentini, Salvatore</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>Executive Vice President</officerTitle>
            <otherText></otherText>
        </reportingOwnerRelationship>
    </reportingOwner>

    <nonDerivativeTable>
        <nonDerivativeTransaction>
            <securityTitle>
                <value>Common Units of limited liability company interests</value>
            </securityTitle>
            <transactionDate>
                <value>2018-11-05</value>
            </transactionDate>
            <deemedExecutionDate></deemedExecutionDate>
            <transactionCoding>
                <transactionFormType>4</transactionFormType>
                <transactionCode>P</transactionCode>
                <equitySwapInvolved>0</equitySwapInvolved>
            </transactionCoding>
            <transactionAmounts>
                <transactionShares>
                    <value>0</value>
                    <footnoteId id="F1"/>
                </transactionShares>
                <transactionPricePerShare>
                    <footnoteId id="F1"/>
                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
                    <value>A</value>
                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>175.88</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. Lentini owns 175.88 Common Units of limited liability company interests (the &quot;Common Units&quot;) of Goldman Sachs Private Middle Market Credit LLC (the &quot;Company&quot;). In addition, on November 5, 2018, the Company issued a capital call notice to its investors that have committed to purchase Common Units. Pursuant to such capital call notice, Mr. Lentini is obligated to make a capital contribution of $1,750 on November 12, 2018 out of his $25,000 aggregate capital commitment to the Company, and the Company is obligated to issue Common Units to Mr. Lentini, provided such capital contribution is funded timely. As of the date of this filing, the number of additional Common Units that will be issued to Mr. Lentini in respect of this $1,750 capital contribution is unknown. Mr. Lentini will amend this Form 4 once the number of Common Units has been determined.</footnote>
    </footnotes>

    <ownerSignature>
        <signatureName>/s/ Maya Teufel, Attorney-in-fact</signatureName>
        <signatureDate>2018-11-07</signatureDate>
    </ownerSignature>
</ownershipDocument>
