Tottenham, Liverpool and Manchester City have all been linked with a move for the 24-year-old.
Boro have already England Under-21 international Adam Johnson and Marvin Emnes to replace Downing.
Downing almost left Boro last year, but decided to stay and signed a new contract with his hometown club.
Reports suggest Middlesbrough boss Gareth Southgate will use some of the money from a sale to launch a bid for Standard Liege youngster Steven Defour.
