Fletcher is a target for the East London club as Hammers boss Sam Allardyce looks to inject more vital experience into his side over the second half of the season.
Manchester United are reportedly battling with the idea of releasing him from his deal early, which would allow West Ham to complete a move for financial reasons.
The paper claims that Fletcher would earn around £1.8 million in wages at United between now and the end of the season.
Should they release the Scot from his current deal, it would mean them receiving a minimal fee and a certain loss against keeping him until the end of the campaign for Louis van Gaal to use as a squad player.
The Guardian claim Allardyce feels the deal is 50/50 at the moment with no resolution likely until heading towards the middle of the week.
West Ham have until the deadline on February 2 to do a deal if any fee is exchanged between the clubs.