The Spain winger enjoyed an impressive loan over the second-half of last season from PSG.
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However, Villa passed on signing Asensio permanently at the end of the campaign after missing Champions League qualification.
French media sources say Villa are now weighing up returning to PSG to setup a new deal for Asensio, who remains available from the European champions.
Villa are making their move as they sell Jacob Ramsey to Newcastle for £40m.