The 29-year-old, who spent the second half of last season on loan with Unai Emery’s team, is regarded as surplus to requirements by the Champions League winners and has been in talks over a possible move to Fenerbahce.
But Asensio has so far been unable to agree personal terms with the Turkish club and his future remains uncertain heading into the final four weeks of the transfer window.
Villa are remaining watchful in the knowledge the Spain international could be made available again on loan nearer the deadline, should he fail to secure a permanent move away.
Asensio scored eight goals during his stay last term and Emery was keen to pursue his signature had Villa managed to qualify for the Champions League.
Missing out on that competition has left the club needing to remain patient as they look to comply with Uefa’s financial fair play rules.
Goalkeeper Marco Bizot has so far been the only senior arrival, with the lack of high-profile departures limiting their ability to spend big to this point.
Moving out fringe players including Alex Moreno and Leander Dendoncker would help bring down the wage bill and give Villa more room to operate.
Exploiting the loan market is one possible strategy with Asensio, Axel Disasi and Marcus Rashford having all joined on a temporary basis in January, though there is an acknowledgement such deals are more likely closer to the deadline.
Villa sold their women’s team in order to comply with the Premier League’s profit and sustainability rules but were fined for breaching the Uefa’s regulations, agreeing a settlement with the governing body which must be followed to avoid further sanction.
Emery has explained how the club must be “more intelligent” in the market due to their lack of financial “power” compared to others.
"We need to join some players in the level we want to achieve, but we don’t have the same power economically like seven other teams in the Premier League,” he said.
“We must be more intelligent and anticipate something. It is very exciting, the work for me and the club. We are really improving and meeting our objectives, but there is still work to do. We have to be demanding and intelligent.”
Villa continue their pre-season schedule when they take on Roma at Walsall’s Pallet Track Bescot Stadium on Wednesday night, just their second and final friendly match on UK soil.