The 28-year-old was put on Newcastle’s transfer list this week after publicly criticising the club’s hierarchy.
Reports have suggested that a number of clubs are interested in signing the midfielder, who has previously been plagued with problems off the field.
“He’s the type of player that, on a free transfer, a lot of clubs in the Premier League will be looking at,” Winterburn told talkSPORT.
“You’ve got to weigh up with Joey Barton whether you can control his temperament, more off the pitch than anything else.
“I think you know that he’s got that competitive edge on the pitch and occasionally it’s going to let him down and it has done before.
“He’s tried to control that, but from Arsene Wenger’s point of view he would need to think that he could control him off the pitch. You can’t have anyone disruptive around the training ground."
The former Manchester City player served a 77-day prison sentence for common assault and affray in Liverpool city centre.
The controversial midfielder was also given a four-month suspended sentence for committing assault occasioning actual bodily harm on his former City teammate Ousmane Dabo.
Barton was fined two weeks’ wages by Newcastle on Tuesday after blasting his reaction to being released on Twitter.