Strachan replaced Gareth Southgate at the Riverside Stadium after Boro failed to push for a swift promotion back to the Premier League.
The former Southampton and Celtic boss was not able to get Boro climbing up the table, though, and with 11 points from as many games at the start of this season increased the pressure on him.
Steve Agnew has been put in charge of Boro's first-team in a caretaker capacity, with his first match a trip to Nottingham Forest on Tuesday evening.
Strachan had two-and-a-half years left on his contract, but has agreed to leave without compensation for the rest of his deal.
Hull City have signed the goalkeeper Vito Mannone on loan from Arsenal until January.
"Vito is a very good young goalkeeper who has found his opportunities limited at Arsenal," said Nigel Pearson, the Tigers manager.
"I like to have competition, in all positions if possible, and Vito has some Premier League and Champions League experience. There'll be some good competition there and it's a healthy position to be in."
Leicester City are in talks with the former Aston Villa forward Darius Vassell, who is a free agent after leaving the Turkish side Ankaragucu in the summer.
Derby County could welcome back midfielder Paul Green against Doncaster on Tuesday, after he missed Saturday's loss to Preston with a hamsring injury.
Matt Timlin, the Portsmouth winger, has extended his loan spell at Swindon Town until November.