Unai Emery would like to sign the striker on a permanent basis following his loan from Paris Saint-Germain last season.
But the deal is a difficult one for Villa, who are working to lower their wage bill to help comply with Uefa’s squad cost rules.
Reports in Turkey say Fenerbahce are ready to pounce for Asensio with their manager, Jose Mourinho, having met with the player to discuss a possible move.
The 29-year-old scored eight goals after joining Villa from PSG in January, helping to reinvigorate the club’s Premier League season alongside fellow loanee Marcus Rashford.
But their failure to qualify again for the Champions League has left Villa facing a tricky summer and while the sale of the women’s team has alleviated immediate pressure posed by PSR, the club face a longer-term struggle to meet Uefa’s requirements.
Next season, clubs will be permitted to spend just per cent of their total revenues on player wages, transfer fees and agents’ fees.
Villa’s percentage is thought to have been above 80 last season and while the punishment for breaching the restrictions at first will amount to a fine, clubs risk more serious penalties if they continue to transgress.