The 23-year-old midfielder has started every match he has been available for this season under Mauricio Pochettino. But his future remains in doubt, with little progress being made to renew his contract with under 18 months left on his current deal.
Cole, who won six major honours in seven years at Chelsea, says Gallagher should snub any interest in January and try to become a legend for his boyhood club.
"Conor Gallagher is a crucial one and is the face of the club," Cole said when launching the Green Football Weekend at Wembley Stadium.
"He's a captain in the leadership group, so to take him out of the team now would not be good. If I was advising him, I'd tell him he has a big future at Chelsea.
"I think what Jordan Henderson was to Liverpool, he can be that for Chelsea."
Chelsea next face Middlesbrough in a crucial Carabao Cup semi-final second leg at Stamford Bridge.
The Blues are 1-0 down after a shock defeat in the first leg, and Cole admits that the match on Tuesday night is key to buying time for Pochettino.
"It's a must-win game because to get knocked out by a Championship side would bring extra heat that's not needed at the moment," he added.
"It's mostly important for the progress of the club and to allow Pochettino to keep doing what he is doing, as getting to another cup final would be epic for these players.
"Since winning the Champions League, we've lost two or three finals and you don't want to become that team.
"We've got to get there first, and Pochettino must keep reminding people to be patient; we're the youngest squad in the Premier League, we need to get players fit, they are also young. So we need to trust in him and his good track record of developing young players."
Cole, who trialled a vegan diet to extend his playing career while at Tampa Bay Rowdies, was involved in a Veggie Cook Off at Wembley ahead of the Green Football Weekend on February 3 and 4.
He is happy to champion a movement which challenges fans to eat one vegetarian meal a day, adding: "Football brings so much to so many people, for the dreams of kids trying to make it here at Wembley to supporters, who want to have their days out.
"Changes could come through using trains instead of planes, eating less meat, and we shouldn't feel guilty about enjoying a game, which has an impact, but just everyone making these small changes will help everyone."