The Gunners and the Blues had both tried to secure the 23-year-old Italian's signature in the summer transfer window, but Roma had remained firm in their decision to hold on to the striker and both the London clubs were forced to sign alternate targets.
Now with Destro reportedly wanting to make a move out of the Serie A side in January due to him getting limited first team opportunities this season, both Arsenal and Chelsea are again back in the hunt.
Roma would also be more than willing to cash in on the striker as Destro, who has managed to net four goals from five league appearances this season, has just been a backup to Francesco Totti and former Arsenal player Gervinho this season.
Jose Mourinho's side, who had also tried to land the striker by offering Fernando Torres in a player plus cash deal this summer, is being heavily linked with a January for Destro, but according to the Daily Star, Arsenal are one step ahead of the Blues as they have already opened informal talks with Roma over a potential transfer for the Italian.
Destro would also be better off moving to Arsenal as he is likely to get more first team chances at the Emirates than Chelsea, who already have the likes of Diego Costa, Didier Drogba and Loic Remy.
Although Arsenal also have the likes of Olivier Giroud, Danny Welbeck, Yaya Sanago and Lukas Podolski to lead their attack, Wenger may be willing to give the young striker a chance as goals have not been easy to come by for Arsenal this season.
Despite Destro being contracted to Roma until 2017, it is believed that a £20 million bid would be enough to land him in January.