Mourinho took over from Pochettino in November but has faced plenty of criticism for his performance to date, overseeing an FA Cup and Champions League last 16 exits in the last month, with the club dropping to eighth in the Premier League table.
"Jose has gone into a difficult situation," Cole said. "In the media in particular there is a lot of anti-Jose, people who are very quick to jump on Jose's back, especially for someone who has been as successful as him.
"You need to look at the job in perspective, I think it had run its course under Pochettino. I think the team wasn't playing well, it needed a change. I think the relationship with Daniel Levy is important, if Levy gets him the players he wants and gives him time I think it will be a success for him."