Cole told The Sun: "Declan is obviously really important. Obviously, there's speculation about his future here, there and everywhere.
"But right now I just don't feel it's the right time for him to be moving on.
"I think he's at a club where they have expectations to push on and I think as long as he understands the project and he knows his place within the team.
"We can't lose too many [players]. We've just lost Mark Noble and I think we've got a few others that we've lost now so we have to bolster and we can't be letting our best players go at the moment."
Cole continued: "If it does happen and someone comes in with an offer that we can't refuse [for Rice] then it's just the way it is.
"We'll reshuffle the pack and nobody in that West Ham squad isn't irreplaceable. You'll always get a few diamonds in the rough and we have found a load of them.
"David Moyes and his scouting network is working tirelessly around the clock right now as we speak to get the best players in for any kind of role that comes up.
"But I can only see Declan Rice's future with us, at least for the current season."