Real Madrid are preparing to offer Asensio, 20, who has a £75million buyout clause a new deal but the Spanish international wants more game time.
He scored stunning goals in both legs of the Spanish Super Cup victory over Barcelona earlier this month but will play second fiddle to Cristiano Ronaldo.
Asensio has already turned down a move to Paris Saint-Germain and is also attracting interest from Inter Milan as well as the Gunners.
The north London club expect to be busy this week with German centre back Shkodran Mustafi a target for both Inter and Juventus.
Arsene Wenger is open to selling Mustafi but wants to recoup as much of the £35million they paid Valencia last summer as possible.
Inter would prefer a season-long loan for the 25 year-old with an option to buy next summer which could be a stumbling block to completing a deal.
Meanwhile Arsenal have agreed a £7million fee with Watford for defender Kieran Gibbs who is keen to make the move to Vicarage Road.
They will have to agree personal terms for Gibbs, 27, who is in the final year of his contract at the Emirates.