The 26-year-old was strongly linked with a move away from the Bernabeu over the summer, but a move did not materialise with the Spain international remaining with Los Blancos.
The attacker is currently down the pecking order in the final third of the pitch.
Barcelona and Manchester United are currently being linked with the Spaniard, but according to Calciomercato, the Gunners and the Reds could make bids during the mid-season transfer window.
The Premier League pair, alongside AC Milan, are also said to be keen to move for the attracter at the beginning of 2023, with the player likely to struggle for regular playtime at Real Madrid throughout the season.
The Spaniard will see his contract at Bernabeu expire at the end of the season, meaning that he would be free to discuss pre-contract terms with foreign clubs in January.
Asensio has contributed 49 goals and 24 assists in 237 appearances for Los Blancos since arriving from Mallorca in 2014.