The Spain international’s situation has been monitored throughout the summer and a return to the Premier League remains possible.
Asensio scored eight goals in 21 appearances while on loan from Paris Saint-Germain during the second half of last season.
Though Villa’s failure to qualify for the Champions League has led to a quiet summer this far, there remains hope a financially-workable deal for Asensio can be reached.
The 29-year-old has only one year remaining on his PSG contract and has been seeking an exit all summer, with a mooted move to Fenerbahce last month failing to materialise.
Though a loan would be Villa’s most likely option, neither has a permanent deal been ruled out.
Villa, meanwhile, are braced for a formal approach from Newcastle for Jacob Ramsey after the midfielder emerged as a possible target for the Magpies.
Ramsey’s future has been the subject of much speculation and both club and player are keeping their options open as the window enters its final weeks.
Talks over a new contract have failed to progress in recent months and Villa are open to selling provided their valuation, thought to be around £40million, is met.
Ramsey’s willingness to leave the club he first joined at the age of six is likely to hinge on the financial package offered by any suitor, along with guarantees over playing time.
The former England under-21 international scored four goals and registered seven assists in 46 total appearances for Villa last season.
Elsewhere, right-back Andres Garcia has emerged as a target for Valencia and could leave Villa just eight months after joining from Levante for £5m.