Allardyce is expected to leave Upton Park this summer having failed to be offered a new deal by the club, with the general consensus being that he will see his tenure brought to a close.
That has led to speculation about who will replace him, with the likes of Jurgen Klopp, Rafa Benitez, Eddie Howe and fans' favourite from his playing days Slaven Billic among those to have been heavily linked with an arrival at the Boleyn Ground - per the Daily Mail.
Cole, though, says the club should target an experienced manager to ensure that they stay in the Premier League before their move to the Olympic Stadium.
“I would go for experience, I wouldn’t risk it and go for an up and coming,” he said.
“I would go for pure experience so you know that you can depend on him to stay in the Premier League. It’s a big season [next season] we need to stay in the Premier League obviously because we’re going to the new Olympic Stadium.”
The striker’s future at the club is also shrouded in doubt, with the 31-year-old also yet to be offered fresh terms to keep him on.
Having spent the last nine years at the club he naturally has an affinity with the Hammers, but Cole has found his playing time limited this season with the likes of Diafra Sakho, Enner Valencia and even injured Andy Carroll ahead of him in the pecking order.
If he were to leave at the end of his current deal, he’s not ruling out a move abroad - should the offer be good enough.
“As a footballer you’ve got to keep your options open,” he added.
“I think if I was to go abroad it wouldn’t be too bad, it would be a new experience for me. I’ve been in the Premier League all my life, to go somewhere else and have a new start would be a new lease of life. If there’s another club, newly promoted or in the Championship, I’d have a look at that too.”