The Daily Telegraph notes that embattled Hammers boss Slaven Bilic has shown interest in Fellaini in the past and could rekindle his interest in the Belgium international to address the club's ongoing problems.
West Ham are just one point above the relegation zone after they were thumped 5-1 at home by Arsenal on 3 December. The Hammers have not won a game in the Premier League since 22 October.
The Telegraph says the West Ham hierarchy is backing Bilic for the moment but the situation could change quickly if results do not improve before Christmas.
Fellaini featured regularly in United's midfield in the early weeks of the season but has been left out of the starting line-up by Jose Mourinho in recent matches.
His performance in United's 1-1 draw with Everton at Goodison Park on 4 December was criticised as "garbage" by Gary Neville.
With United defending a 1-0 lead courtesy of Zlatan Ibrahimovic's first-half goal, Fellaini came on as a substitute in place of Henrikh Mkhitaryan in the 85th minute and conceded a penalty by fouling Idrissa Gueye, allowing Leighton Baines to equalise from the spot.
"It's just rubbish from Fellaini. It's obvious – as soon as you commit yourself in the box you're in danger of giving a penalty away. It's garbage from him," the former Manchester United defender was quoted as saying by the Sun.
"[His introduction] was purely to deal with the long balls into the box. You can see the sense in it but when he goes on and does such an idiotic thing like that. It was really poor from an experienced international."
West Ham have also been linked with a move for Chelsea goalkeeper Asmir Begovic, who is growing increasingly frustrated at his lack of game time under Antonio Conte, according to the Sun.
The club are also reportedly looking it at bringing in a new right-back and a striker.