The 23-year-old has been at the Stadio Olimpico since 2012, in which time he has built on his reputation as one of Serie A’s top young forwards.
Despite averaging almost a goal every other game in the red shirt, Roma are said to be open to a deal.
The club’s director Walter Sabatini is understood to have travelled to England with the intention of assessing the interest in his player.
Spurs are known to have a good relationship with the Rome giants, with Franco Baldini having worked for them before joining the Lilywhites in a behind the scenes role.
A new striker is thought to be tempting for Mauricio Pochettino, who is rumoured to be unconvinced by Roberto Soldado.
Destro has been capped five times by Italy, scoring one goal in the process.