The 28-year-old has reignited his career since moving to West Ham United on loan in January, chipping in with eight goals and four assists in nine Premier League appearances for David Moyes's Champions League hopefuls.
A source at West Ham has affirmed that the club will do "whatever it takes" to land Lingard on a permanent deal this summer, and Cole believes that the time is right for United to cash in on the England international.
Speaking to talkSPORT Cole said: "The opportunity for him to go on loan to West Ham came at a fantastic time. He's taken it with both hands.
"Now Manchester United have a decision to make - [do they] take him back or sell him in the summer for a little bit more money than was expected?
"I think if I was Manchester United, I would move him on. And if I was Jesse I would want to go and play games. Definitely."