Adomah is keen to leave Ashton Gate following City's relegation to League One and he saw a transfer request refused earlier this summer.
Wigan Athletic have had two bids turned down and City boss Sean O'Driscoll stated last month that he was keen for a swift resolution to the situation.
It now appears that Adomah is close to finalising a move and he could become Middlesbrough's fourth signing of the summer.
Boro boss Tony Mowbray confirmed on Monday that he has offers in for several players as he tries to further strengthen his squad before the transfer window closes.
He said in the Gazette: "My job is to try and identify footballers who can try to help the team. It's not my job to try to negotiate transfers so I don't know where that's at.
"I would hope we will have two or three players in in the next week or two which will freshen the team up to the point where it will create more competition."
Mowbray added: "We've been trying for a few weeks now and it's not easy when you haven't got a centre-forward on the bench who thinks he should be playing and is ready to rip his tracksuit off and come on and show you that he should have been starting.
"But hopefully we'll rectify that and create some competition in the next few weeks.
"We will try to strengthen in a few areas of the pitch before the window shuts. We've got two or three bids in for different players and we wait for those clubs to make decisions on their futures."