The former Arsenal star has entered the final 12 months of his contract at the Hammers.
He is also struggling to convince David Moyes of his worth after enduring all too familiar injury struggles while in the East End.
It has been suggested that he could be released by the Hammers, with other options being favoured.
Cole feels an American adventure should appeal to Wilshere if another door opens for him, with the ex-England international having spent time in the States with NASL side Tampa Bay Rowdies towards the end of his career.
“I really like Jack, he’s a fantastic player and I think he’s very similar to myself, in that when he came onto the scene it was all very exciting for everyone,” former West Ham midfielder Cole told talkSPORT.
“I feel for Jack because he hasn’t even been able to really get there, he’s never really played regularly in his career and he’s 28 now.
“His injuries have been so much, I think he just need to play football now and enjoy himself.
“He’s got a young family and he would absolutely love the States, particularly if he goes to a place like Miami or Tampa, somewhere warm.
“It actually makes your body feel a lot better playing in those climates.
“I’m a big fan of Jack’s and I’d just love to see him playing football, that’s his natural habitat and he’s such a talented player.”
Wilshere is eager to continue testing himself at the highest level and has hinted that he will look to stay in Europe if he severs ties with West Ham.
He told The Times: “I just want to play games and get back to the level whether that's at West Ham or somewhere else, different country, a different culture maybe.
“I want to prove people wrong and most of all for myself to see if I can get myself to a high level again.
“I want to show people what I can do. Any footballer that says he's happy being on the bench is lying. I want to play.
“I want to get a run of games, 15-20 games which I still think I can do and have an influence on games and get that confidence back.”