The 30-year-old, who left the Riverside two seasons ago while in the Premier League, has been increasingly linked to the club after catching the eye of manager Tony Pulis following his 14 goals and five assists in 37 championship games last season.
Former Crystal Palace chairman Simon Jordan has urged Villa to sell Adomah, Jonathan Kodjia and Jack Grealish to reduce their wage bill.
“They’ve got some key assets, in Grealish, Adomah and possibly Kodjia who could be sold," he told Talk Sports.
“I’m not suggesting they should sell their assets to consolidate. I’m suggesting they could sell two assets, capitalise on them at a decent price, to be able to make their squad more competitive and then go again next season,” he added.
Goal understands that Leeds United have also joined the race for Adomah and could step in if Middlesbrough's deal falls through.