Winger Tavernier, 18, has established himself in the Boro team since making his senior debut in the EFL Cup win against Scunthorpe in August.
He hit the winner against Sunderland in the Tees-Wear derby before representing England at Under-19 level during the international break.
Premier League giants Arsenal have been alerted to his rise, sending scouts to track his performances.
Everton are also interested and could make a bid in the January transfer window, according to the Daily Mail.
The Toffees have a track record of signing promising players from lower-league clubs in recent seasons, having snapped up John Stones, Dominic Calvert-Lewin and Ademola Lookman.
Bournemouth and Middlesbrough’s Championship rivals Aston Villa are also reportedly fans of Tavernier.
Tavernier’s contract at the Riverside Stadium expires next summer.
Boro chiefs, however, are set to trigger a one-year option.
Middlesbrough veteran Stewart Downing has been impressed by Tavernier’s impressive performances.
“Marcus probably thinks it’s easy at the moment, doesn’t he,” Downing said.
“Two goals, four wins and he’s riding high. The [Sunderland] derby is a big game but he played with no fear and really handled it.
“The lad has been a breath of fresh air. He got his chance due to injuries and he’s really grabbed it.
“He’s only 18 and it won’t be plain sailing all the time but let him enjoy it for as long as he can.”