The 34-year-old Dyer is currently a free agent after being released by Queens Park Rangers earlier this month.
"I've had some conversations with Kieron," said Mowbray. "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."
Dyer has been training with Ipswich Town in recent days in a bid to maintain his fitness.