The central defender has not played for the club since last March when he suffered a fifth cruciate ligament injury.
His contract expired in June, but Bates was allowed to continue his rehabilitation with the Teessiders and is now fit to return.
But he will have to find a new club after terms could not be agreed in meetings this week.
Middlesbrough boss Tony Mowbray said the split was amicable.
"It was almost a natural parting of the ways," he said.
"We've allowed him to get himself fit. He got injured under contract and the right thing to do was to allow Matthew full rehabilitation, all the nutrition that goes with it.
"He's at the stage now where he probably needs to play a football match. We had talks on a contract, and we were both quite a way apart, and ultimately the decision was made that Matthew would move on. We wish him well"
Mowbray also hinted that last season's club captain could have found himself surplus to requirements at the Riverside.
"When he got injured he was captain of our football club and a main player, and yet things have moved on.
"My job was to make sure we had enough centre-halves. We have new players like Jonathan Woodgate, George Friend and Andre Bikey, and we have Seb Hines [an academy team-mate of Bates'] - we have a lot of cover.
"It's almost difficult in my mind for Matthew to come back into our club as number four or number five centre-half."
Chief executive Neil Bausor paid tribute to the talented defender.
He said: "After discussions with Matthew, all parties agreed that it would be best if he moved on.
"It was an amicable meeting and we wish Matthew all the best for the future. He's been a great servant for Middlesbrough Football Club."