The 60-year-old former Bolton Wanderers boss is out of contract in the summer and will not hold discussions with the Hammers' hierarchy until the end of the season.
Fellow-strugglers Sunderland have already attempted to lure 'Big Sam' to the North East as a long-term replacement for Gus Poyet who was sacked last month.
According to the Daily Mirror, current R's boss Chris Ramsey has the support of director of football Les Ferdinand but lacks the experience of Allardyce.
Mark Warburton, who is expected to leave Brentford in the summer, has also been linked with the job but has no Premier League know-how.
The Hoops' players have reportedly responded positively to Ramsey since he was put in charge at the start of February - despite being in the relegation zone.
Six defeats in seven outings since Harry Redknapp’s departure have increased fears of returning to the Championship.
If that were to happen, it remains to be seen if Allardyce would be interested in the post.