The Hammers man is one of a number of players who is looking to leave the Europa Conference League winners this summer.
Scamacca has made 27 appearances since signing for the east London club last summer, scoring eight goals as they swerved away from Premier League relegation.
But the Italian will have to wait to find his exit route now West Ham have declined Inter's offer.
Meanwhile, the Hammers are ready to walk away from their interest in Southampton midfielder James Ward-Prowse.
That's despite the Hammers looking for a replacement for Declan Rice, who was sold to London rivals Arsenal for £105m earlier this month - not to mention having a healthy transfer budget as a result.
West Ham's have also been unsuccessful in their pursuit of Man United duo Harry Maguire and Scott McTominay so far.
The Hammers' £20m bid for Maguire was knocked back by the Red Devils last week. Meanwhile, Man United slapped a £40m price tag on McTominay.
David Moyes's side haven't made a signing in 47 days, bringing in just one player this summer - 17-year-old Sean Moore from Northern Irish club Cliftonville for an undisclosed fee.