<?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>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>2017-05-19</value>
            </transactionDate>
            <deemedExecutionDate></deemedExecutionDate>
            <transactionCoding>
                <transactionFormType>4</transactionFormType>
                <transactionCode>P</transactionCode>
                <equitySwapInvolved>0</equitySwapInvolved>
            </transactionCoding>
            <transactionAmounts>
                <transactionShares>
                    <value>80.94</value>
                </transactionShares>
                <transactionPricePerShare>
                    <value>98.8336</value>
                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
                    <value>A</value>
                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>508.55</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 indirectly holds 427.61 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. Kari is obligated to make a capital contribution of $8,000 on May 26, 2017 out of his $200,000 aggregate capital commitment to the Company, and the Company is obligated to issue approximately 80.94 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/ Neena A. Reddy, Attorney-in-fact</signatureName>
        <signatureDate>2017-05-23</signatureDate>
    </ownerSignature>
</ownershipDocument>
