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

    <schemaVersion>X0206</schemaVersion>

    <documentType>3</documentType>

    <periodOfReport>2016-09-11</periodOfReport>

    <noSecuritiesOwned>0</noSecuritiesOwned>

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

    <reportingOwner>
        <reportingOwnerId>
            <rptOwnerCik>0001635549</rptOwnerCik>
            <rptOwnerName>Yoder, Jon</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>Chief Operating Officer</officerTitle>
            <otherText></otherText>
        </reportingOwnerRelationship>
    </reportingOwner>

    <nonDerivativeTable>
        <nonDerivativeHolding>
            <securityTitle>
                <value>Common Units of limited liability company interests</value>
            </securityTitle>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>30.19</value>
                    <footnoteId id="F1"/>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </nonDerivativeHolding>
    </nonDerivativeTable>

    <derivativeTable></derivativeTable>

    <footnotes>
        <footnote id="F1">On the date hereof, Mr. Yoder owns 30.19 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 September 8, 2016, the Company issued a capital call notice to its holders of Common Units. Pursuant to such capital call notice, Mr. Yoder is obligated to make a capital contribution of $7,500 on September 15, 2016 out of his $50,000 aggregate capital commitment to the Company, and the Company is obligated to issue Common Units to Mr. Yoder, 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. Yoder in respect of this $7,500 capital contribution is unknown.</footnote>
    </footnotes>

    <remarks>Exhibit List: Exhibit 24 - Power of Attorney</remarks>

    <ownerSignature>
        <signatureName>/s/ Kathryn Quirk, Attorney-in-fact</signatureName>
        <signatureDate>2016-09-12</signatureDate>
    </ownerSignature>
</ownershipDocument>
