The Riverside Stadium outfit are in the market for a fresh face in midfield, and the Boro coach is convinced the former QPR man will be a vital player should he make the move
Recently released from his contract at QPR, the 34-year-old has been training with former club Ipswich in order to maintain fitness but is unlikely to re-join the Championship outfit.
Dyer has struggled with injuries in recent years, having only made five first-team for the Loftus Road outfit since joining them in 2011.
Mowbray however, has hinted that the former England international could have a place at the Riverside Stadium and would prove to be "a huge asset" for the club if he stayed fit.
"I've had some conversations with Kieron," Mowbray told Sky Sports. "If Kieron was fit and playing for us then he would be a huge asset for our club.
"He's scored in the FA Cup against West Brom and he's been available, fit and playing. I'll have to see if something can be done and we might be close to that."
Middlesbrough have struggled with consistency in recent weeks, but Mowbray is hopeful the addition of Dyer could help his side mount a bid for promotion come the end of the season.
"I've watched some footage of him playing against Gareth Bale for QPR and he did well," he added.
"If Kieron plays football for us and helps win football matches then I think he would be an asset."