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

    <schemaVersion>X0306</schemaVersion>

    <documentType>4</documentType>

    <periodOfReport>2017-04-20</periodOfReport>

    <notSubjectToSection16>0</notSubjectToSection16>

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

    <reportingOwner>
        <reportingOwnerId>
            <rptOwnerCik>0001330118</rptOwnerCik>
            <rptOwnerName>Kari Ross Jay</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>1</isDirector>
            <isOfficer>0</isOfficer>
            <isTenPercentOwner>0</isTenPercentOwner>
            <isOther>0</isOther>
            <officerTitle></officerTitle>
            <otherText></otherText>
        </reportingOwnerRelationship>
    </reportingOwner>

    <nonDerivativeTable>
        <nonDerivativeTransaction>
            <securityTitle>
                <value>Common Units of limited liability company interests</value>
            </securityTitle>
            <transactionDate>
                <value>2017-04-20</value>
            </transactionDate>
            <deemedExecutionDate></deemedExecutionDate>
            <transactionCoding>
                <transactionFormType>4</transactionFormType>
                <transactionCode>P</transactionCode>
                <equitySwapInvolved>0</equitySwapInvolved>
            </transactionCoding>
            <transactionAmounts>
                <transactionShares>
                    <value>427.61</value>
                </transactionShares>
                <transactionPricePerShare>
                    <value>98.2194</value>
                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
                    <value>A</value>
                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>427.61</value>
                    <footnoteId id="F1"/>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>I</value>
                </directOrIndirectOwnership>
                <natureOfOwnership>
                    <value>By The Kari Family Trust U/A/D 6/15/99</value>
                    <footnoteId id="F2"/>
                </natureOfOwnership>
            </ownershipNature>
        </nonDerivativeTransaction>
    </nonDerivativeTable>

    <derivativeTable></derivativeTable>

    <footnotes>
        <footnote id="F1">On the date hereof, Mr. Kari does not own any Common Units of limited liability company interests (the &quot;Common Units&quot;) of Goldman Sachs Private Middle Market Credit LLC (the &quot;Company&quot;). However, on April 20, 2017, the Company issued a capital call notice to certain investors that have committed to purchase Common Units. Pursuant to such capital call notice, Mr. Kari is obligated to make a capital contribution of $42,000 on April 27, 2017 out of his $200,000 aggregate capital commitment to the Company, and the Company is obligated to issue approximately 427.61 Common Units to Mr. Kari, provided such capital contribution is funded timely.</footnote>
        <footnote id="F2">These shares are held in a trust for the benefit of the reporting person's family members. The reporting person and the reporting person's spouse are trustees of the trust. The reporting person disclaims beneficial ownership of these securities, and the filing of this report is not an admission that the reporting person is the beneficial owner of these securities for purposes of Section 16 or for any other purpose.</footnote>
    </footnotes>

    <ownerSignature>
        <signatureName>/s/ Judith Shandling, Attorney-in-fact</signatureName>
        <signatureDate>2017-04-24</signatureDate>
    </ownerSignature>
</ownershipDocument>
