QPR loanee Tom Carroll wants to "make it at Tottenham" but is prepared to consider all of his options at the end of the season.
Ilkay Gundogan has commited his future to Borussia Dortmund by signing a new contract.
The Boro boss was speaking after his side edged past League Two side Aldershot 2-1 in their FA Cup fourth-round tie at the Riverside.
Dyer was recently released by QPR and Mowbray said: "I've had some conversations with Kieron. If Kieron was fit and playing for us then he would be a huge asset for our club.
"I've watched some footage of him playing against Gareth Bale for QPR and he did well. 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.
"If Kieron plays football for us and helps win football matches then I think he would be an asset."