The Roma-owned striker is currently playing in Turkey on-loan with Besiktas.
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The Times says Villa want to sign Abraham permanently this month and have tabled a contract to 2030 to the England international.
Abraham had a superb previous spell on-loan at Villa Park while a Chelsea player.
Villa boss Unai Emery is eager to see the deal happen and is prepared to sacrifice Donyell Malen, who is set to join Roma, and Evann Guessand to smooth negotiations.
Villa are willing to send Guessand on-loan to Besitkas to soften the blow of losing Abraham at this stage of the season.