The 20-year-old is manager Steve Bruce's top transfer target and has been all summer, according to the Birmingham Live's Gregg Evans.
However, Aston Villa face competition for Abraham's signature. There is also interest in the young striker from clubs in Italy and the Netherlands, per the report.
New Chelsea manager Maurizio Sarri has been busy trimming his squad by sending some of the club's fringe players out on loan. Kurt Zouma, Michy Batshuayi and Tiemoue Bakayoko have all departed this summer.
Sarri has explained what he wants from his squad:
It would not be a big surprise if Abraham were another player deemed surplus to requirements by the new manager. He spent last season on loan at Swansea City, where he managed eight goals in 41 appearances.
He looks unlikely to see much game time at Stamford Bridge, as he is behind both Alvaro Morata and Olivier Giroud in the pecking order.
Darren Lewis at the Mirror said there has been plenty of interest in Abraham:
Aston Villa will be hoping to go one better than last season when they narrowly missed out on promotion to the Premier League. The Villains were beaten 1-0 by Fulham in the Championship play-off final.
The club have shown their intent over the summer by managing to keep hold of key midfielder Jack Grealish despite receiving a £25 million bid from Tottenham Hotspur, per James Dale at Sky Sports News.
Aston Villa announced in July they had received "significant investment capital" from NSWE, a company owned by Nassef Sawiris and Wes Edens, per the club's official website.
The investment means Villa have no need to sell their best players and also have funds available for new signings.
The club are said to want to bring in Abraham on loan initially with a view to signing him permanently, per the report.
It remains to be seen if Abraham would be willing to drop down a division. However, a move to Aston Villa would allow him regular game time and the chance to develop his career.
Chelsea may also need some convincing to let him depart having already allowed Batshuayi to join Valencia. An injury to either Morata or Giroud would leave Chelsea short of striking options.