The No.1 role at Stamford Bridge has come under increased pressure in recent years, with both Robert Sanchez and Filip Jorgensen struggling to make their mark under several managers.
Pickford, who has been capped 83 times for his country and will play a huge role in the upcoming World Cup, has been England’s No.1 goalkeeper for almost a decade.
Speaking to BOYLE Sports, who offer the latest World Cup odds, Sheringham suggested that the Blues make an audacious move for Pickford in what would be one of the biggest moves of the summer.
"For a goalkeeper, Jordan Pickford is still young. He’s right in his prime. Let’s not forget Peter Schmeichel and David Seaman kept going on until 38 and 39 and into their 40s. When you're good, you're good, and you know it.
"So there's still time for Pickford to, dare I say this to the Everton fans, to a bigger club in the top six, competing for the big titles. There is still time for him to do that, but he's obviously enjoying what he's doing now, and I’m sure it’s the question that’s been put to him time and time again over the last few years.
"If he wanted to go, I'm sure David Moyes wouldn't stand in his way because of the service that he has given Everton, and I would understand if he wanted to go on and further his career.
"Obviously Moyes wants to take Everton to the next stage too so he wouldn't want that, but I think it's all about what Pickford wants, really. Number one playing for England but not playing for a top club? I know that will infuriate Everton fans to hear me say that.
"Manchester United don’t need a new goalkeeper now. They’re happy with the one they’ve found, Senne Lammens. Chelsea need a top class goalkeeper, and Pickford would tick every box for what they’re looking for, without a doubt."
Everton finished four points off the final European spot in the Premier League with the Europa Conference League qualification going to eighth place. Pickford may be looking for more success as he considers his future and Chelsea may be the team to step up.