<?xml version="1.0"?><ownershipDocument><schemaVersion>X0101</schemaVersion>    <documentType>4</documentType><periodOfReport>2003-06-18</periodOfReport><issuer><issuerCik>0001054721</issuerCik><issuerName>BSQUARE CORP /WA</issuerName>        <issuerTradingSymbol>BSQR</issuerTradingSymbol></issuer><reportingOwner><reportingOwnerId><rptOwnerCik>0001242316</rptOwnerCik><rptOwnerName>ASP NOGI</rptOwnerName></reportingOwnerId><reportingOwnerAddress><rptOwnerStreet1>  </rptOwnerStreet1><rptOwnerStreet2>  </rptOwnerStreet2><rptOwnerCity>  </rptOwnerCity><rptOwnerState>  </rptOwnerState><rptOwnerZipCode>  </rptOwnerZipCode></reportingOwnerAddress><reportingOwnerRelationship><isDirector>0</isDirector><isOfficer>1</isOfficer><isTenPercentOwner>0</isTenPercentOwner><isOther>0</isOther><officerTitle>Vice Pres. Finance, CFO</officerTitle><otherText></otherText></reportingOwnerRelationship></reportingOwner><derivativeSecurity><securityTitle><value>Employee Stock Option</value></securityTitle><conversionOrExercisePrice><value>0.91</value></conversionOrExercisePrice><transactionDate><value>2003-06-16</value></transactionDate><transactionCoding><transactionFormType>4</transactionFormType><transactionCode>A</transactionCode><equitySwapInvolved>0</equitySwapInvolved></transactionCoding><transactionTimeliness><value></value></transactionTimeliness><transactionAmounts><transactionShares><value>75000</value></transactionShares><transactionValue><value>0</value></transactionValue><transactionAcquiredDisposedCode><value>A</value></transactionAcquiredDisposedCode></transactionAmounts><exerciseDate><value>2003-09-16</value><footnoteId id="F1" /></exerciseDate><expirationDate><value>2013-06-16</value></expirationDate><underlyingSecurity><underlyingSecurityTitle><value>Common Stock</value></underlyingSecurityTitle><underlyingSecurityShares><value>75000</value></underlyingSecurityShares></underlyingSecurity><postTransactionAmounts><sharesOwnedFollowingTransaction><value>75000</value></sharesOwnedFollowingTransaction></postTransactionAmounts><ownershipNature><directOrIndirectOwnership><value>D</value></directOrIndirectOwnership></ownershipNature></derivativeSecurity><footnotes><footnote id="F1">Grant vests 1/8 on the date indicated as "date exercisable" and 1/8 every three months thereafter until fully vested</footnote></footnotes><ownerSignature><signatureName>Nogi Asp</signatureName><signatureDate>2003-06-18</signatureDate></ownerSignature></ownershipDocument>

