It is thought that the Giallorossi would be willing to allow Destro to move on, having loaned him out to AC Milan for the second half of last season.
He joined Roma from Siena in 2012 and has gone on to score 25 goals in 57 league appearances for them.
According to Daily Mail, Norwich have made a formal £4.5m offer for Destro.
West Ham United and Bologna have also been linked with the attacker, who netted three goals in his 15 games at Milan last term.
Destro has won eight caps for Italy, scoring one goal.