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

    <schemaVersion>X0206</schemaVersion>

    <documentType>3</documentType>

    <periodOfReport>2024-03-15</periodOfReport>

    <noSecuritiesOwned>1</noSecuritiesOwned>

    <issuer>
        <issuerCik>0001069202</issuerCik>
        <issuerName>LENNOX INTERNATIONAL INC</issuerName>
        <issuerTradingSymbol>LII</issuerTradingSymbol>
    </issuer>

    <reportingOwner>
        <reportingOwnerId>
            <rptOwnerCik>0001424543</rptOwnerCik>
            <rptOwnerName>Somasundaram Sivasankaran</rptOwnerName>
        </reportingOwnerId>
        <reportingOwnerAddress>
            <rptOwnerStreet1>2140 LAKE PARK BLVD.</rptOwnerStreet1>
            <rptOwnerStreet2>8TH FL</rptOwnerStreet2>
            <rptOwnerCity>RICHARDSON</rptOwnerCity>
            <rptOwnerState>TX</rptOwnerState>
            <rptOwnerZipCode>75080</rptOwnerZipCode>
            <rptOwnerStateDescription></rptOwnerStateDescription>
        </reportingOwnerAddress>
        <reportingOwnerRelationship>
            <isDirector>1</isDirector>
        </reportingOwnerRelationship>
    </reportingOwner>

    <footnotes></footnotes>

    <remarks>Attorney-in-fact pursuant to power of attorney dated March 15, 2024.</remarks>

    <ownerSignature>
        <signatureName>/s/ Monica M. Brown, attorney-in-fact for Mr. Sivasankaran Somasundaram</signatureName>
        <signatureDate>2024-03-18</signatureDate>
    </ownerSignature>
</ownershipDocument>
