The 34-year-old is a free agent following his release by QPR earlier this month, and Mowbray believes the former England midfielder would be a valuable asset in their promotion push this season.
Mowbray is keen to have the deal in place in time for Dyer to face his first club Ipswich at Portman Road this weekend.
"It's a deal that suits both parties," said Mowbray. "Kieron and his advisors have been very, very honourable in the discussions.
"I'm happy he's here. He's a good lad and a good pro. He wants to play football and he's happy to be here - hopefully he can contribute and help us succeed.
"I've known Kieron a long time and coached him for a year or two at Ipswich before he moved on. At such a young age, he was a fabulous footballer and he's still a fabulous footballer.
"Obviously he's had injury problems for a lot of years. People keep throwing statistics at me of how many games he's played and for how long, but I look at his record of where he has played this year.
"I watched him play against Manchester United this season and against Gareth Bale at Tottenham, and he was fine. We will have to see how he handles the intensity required in this league."
Meanwhile, Middlesbrough confirmed on Thursday night that Kevin Thomson has had his contract cancelled by mutual consent, while 19-year-old forward Jake Fowler has joined Stockport on a one-month deal.