Everton have been linked heavily with Hart this summer (The Sun), with the Merseyside club on the look out for a new stopper.
Everton have already snapped up Marteen Stekelenburg, but they are expected to invest further funds in a new number one, with Hart on their radar.
Hart’s future at City is in doubt, following the appointment of Pep Guardiola.
There have been suggestions that City could make a move for Marc Andre Ter Stegen, as Guardiola seeks a goalkeeper who is more comfortable with the ball at his feet.
However, Given, who played with Hart for a stage of his career at City, suggests that the England international will want to battle it out to prove his worth to Guardiola.
“It depends if they bring someone in [whether he’d move to Everton]. I think Joe will want to go and fight,” Given said.
“If Guardiola brings someone in, I think he’d definitely get itchy feet.”
Hart’s reputation has taken a battering this summer, after a disastrous Euro 2016.
Hart was at fault for two of the goals England conceded at the tournament, with one high-profile error in the Three Lions’ 2-1 defeat to Iceland drawing heavy criticism.