Taylor, 24, emerged from the club's Academy as a left-back and that is the position in which he has played largely ever since.
However, he has lined up in midfield for Boro since his return from a loan spell at Watford, who have offered him a permanent deal, and with youngster Joe Bennett having been handed a new contract earlier this week, Mowbray wants the older man to consider making the change for good.
The manager told the club's official website: "My job is to try and persuade Andrew that his future is with Middlesbrough. He has great ball retention and a football brain.
"I think he can be a high-class midfielder, which means he's got some thinking to do because I know he has previously seen himself only as a left-back because that is where he has played most of his football.
"Ultimately, whether he stays or goes at this point is my decision because he is our footballer. The boy is trying to make a footballing decision and I'm trying to convince him that he can make a top-class midfield player, where possibly he can take his career to a new level."