The Hammers midfielder is being linked with United this season.
Cole told BT Sport: "I wouldn't go if I was Declan Rice, no. I think you run the risk of going there and still not winning trophies.
"I think if Man City were on the table, you'd go there because in recent years it guarantees you trophies and I think Declan would want that.
"But for me there's no reason for him to not stay at West Ham if the progress is still going in that [upward] pattern, but that's for him and the owners.
"He needs better players around him at West Ham to go to that next level."
Need a VPN? Or seeking to switch? Then support Tribal Football by clicking here to sign up with our official partners NordVPN. Not only will you receive the best VPN experience, but also great discounts by signing up via Tribal Football. Thanks for your support!