The player-turned-pundit is of the view that the England international must leave Manchester United in January after falling out of favour under Ruben Amorim.
He feels a team abroad or a “progressive” club in the Premier League would be suited to the forward, with the Irons among those mentioned.
“Marcus Rashford should leave Manchester United”, Collymore wrote in his CaughtOffside column (25 December).
“I’ve been saying it for probably a couple of years now, and he should either go abroad or to a progressive Premier League club.
“Would that be a West Ham or a Brighton or would it even be a Bournemouth?”
Could West Ham pursue a deal for Marcus Rashford?
West Ham are in need of a new centre-forward, with Michail Antonio, Niclas Fullkrug and Danny Ings in their 30s.
Antonio’s unfortunate road traffic accident has also left Julen Lopetegui without his first-choice striker, meaning the Hammers could enter the market for a new one in the January transfer window.
Rashford, meanwhile, is facing an uncertain future at Manchester United after he was excluded from each of their last three matchday squads.
Therefore, the East London outfit have a chance to sign the 27-year-old if they want, having also done business with the Red Devils for Aaron Wan-Bissaka’s signing in the summer.
However, Rashford’s wages are likely to prove prohibitive.
West Ham are thus better off looking elsewhere, with several players already linked with a move to the London Stadium.
Mainz hitman Jonathan Burkardt is said to be on Tim Steidten’s transfer wishlist ahead of January, with Brighton’s Evan Ferguson also an option [TBR Football].