The 32-year-old was a second-half substitute during Sunday's FA Cup win over Sheffield Wednesday.
It was Downing's 19th appearance of the season but he has not been a regular starter for Aitor Karanka's side.
He rejoined his home-town team in July 2015 on a four-year deal and since played 68 times, scoring four goals.
Downing, who played 234 times for Middlesbrough in his initial spell before joining Aston Villa for £12m in July 2009, has scored once this season in October's 2-0 Premier League win over Bournemouth.
Former England winger Stewart Downing will be allowed to leave Middlesbrough on loan during the January transfer window, BBC Tees reports.