The England international has been given a chance to kickstart his career after falling down the pecking order at Old Trafford.
ole told BT Sport: "I've always been a fan of Jesse. He's lost his way in the last few years.
"Some fault will go to the managers not fancying him but this is what he brings to this club.
"He's found a club where they'll love a player like him - hard-working, hustle and bustle, but also with quality and the ability.
"David alluded to the fact that when he was at Manchester United as a youngster, he saw the talent, and that sticks with you as a manager.
"He's going to be looking at him and thinking: 'I can get him back to his best'.
"And his very best was playing number 10 for England. We must not forget that but he's 28 now, he needs to play. He hasn't played enough games in his career, so he may not even be at his peak.
"He's at the right club, he's got to look at the players around him and think: 'We can do something here'."
On the possibility of Lingard's move becoming permanent, Cole added: "Listen, he'll come and do his job and that will make West Ham want to buy him, I'm pretty sure.
"Then you get into the issues of transfer fees and all that but if you're going to be serious now, you've got a player in the door who can do a job for you, get the deal done, get him here and for Jesse it would be great as well."