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

    <schemaVersion>X0508</schemaVersion>

    <documentType>4</documentType>

    <periodOfReport>2025-01-02</periodOfReport>

    <notSubjectToSection16>0</notSubjectToSection16>

    <issuer>
        <issuerCik>0001838406</issuerCik>
        <issuerName>BKV Corp</issuerName>
        <issuerTradingSymbol>BKV</issuerTradingSymbol>
    </issuer>

    <reportingOwner>
        <reportingOwnerId>
            <rptOwnerCik>0001953916</rptOwnerCik>
            <rptOwnerName>Larrick Lindsay B</rptOwnerName>
        </reportingOwnerId>
        <reportingOwnerAddress>
            <rptOwnerStreet1>1200 17TH STREET, SUITE 2100</rptOwnerStreet1>
            <rptOwnerStreet2></rptOwnerStreet2>
            <rptOwnerCity>DENVER</rptOwnerCity>
            <rptOwnerState>CO</rptOwnerState>
            <rptOwnerZipCode>80202</rptOwnerZipCode>
            <rptOwnerStateDescription></rptOwnerStateDescription>
        </reportingOwnerAddress>
        <reportingOwnerRelationship>
            <isDirector>0</isDirector>
            <isOfficer>1</isOfficer>
            <isTenPercentOwner>0</isTenPercentOwner>
            <isOther>0</isOther>
            <officerTitle>Chief Legal Officer</officerTitle>
            <otherText></otherText>
        </reportingOwnerRelationship>
    </reportingOwner>

    <aff10b5One>0</aff10b5One>

    <nonDerivativeTable>
        <nonDerivativeTransaction>
            <securityTitle>
                <value>Common Stock</value>
            </securityTitle>
            <transactionDate>
                <value>2025-01-02</value>
            </transactionDate>
            <transactionCoding>
                <transactionFormType>4</transactionFormType>
                <transactionCode>F</transactionCode>
                <equitySwapInvolved>0</equitySwapInvolved>
            </transactionCoding>
            <transactionAmounts>
                <transactionShares>
                    <value>2518</value>
                    <footnoteId id="F1"/>
                </transactionShares>
                <transactionPricePerShare>
                    <value>23.78</value>
                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
                    <value>D</value>
                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>234756</value>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </nonDerivativeTransaction>
    </nonDerivativeTable>

    <derivativeTable></derivativeTable>

    <footnotes>
        <footnote id="F1">Represents shares of common stock, par value $0.01 per share (&quot;Common Stock&quot;), withheld to satisfy tax withholding obligations upon the vesting of restricted stock units previously reported on Form 4. This net settlement was approved by the board of directors of the Issuer pursuant to Rule 16b-3 under the Securities Exchange Act of 1934, as amended.</footnote>
    </footnotes>

    <remarks></remarks>

    <ownerSignature>
        <signatureName>/s/ Kathleen Lenox, attorney-in-fact</signatureName>
        <signatureDate>2025-01-03</signatureDate>
    </ownerSignature>
</ownershipDocument>
