The Express & Star understands that the 21-year-old is edging nearer to joining on a temporary basis for the rest of the season – with no option to buy him permanently.
Chelsea are thought to be reluctant to sell the England international striker.
Wolves believe Abraham, who has netted 16 times in 20 appearances for Villa, can help them secure a top-half finish this season and help take the load of Raul Jimenez, who is the only out-and-out striker to play for them in the last two months with Leo Bonatini out of favour.
There had been suggestions that Abraham wouldn't be able to move to Molineux as he'd already played for Villa and Chelsea this season.
However Wolves had already done their due diligence before getting into negotiations with the Blues. Abraham had featured for them in the Community Shield and an Under-23 game but neither pose a problem to the deal being completed.
Chelsea have until January 14 to trigger a recall clause in Abraham’s loan agreement. Wolves will likely be paying a large proportion if not all of Abraham's £55,000 weekly wage.
Wolves head coach Nuno Espirito Santo remained coy on the issue yesterday, saying he "didn't know" if Wolves had been speaking to Chelsea.
Villa boss Dean said: "I’ve told him to go with his gut feeling. If it’s to stay at Aston Villa and become a legend – because he could be the one who fires us back to the Premier League – then I would be only too happy."