David Moyes' men have shown some much more encouraging signs in recent weeks, beating Chelsea and the Potters and securing draws against Arsenal and Bournemouth, but they still find themselves in a very precarious position in the Premier League.
The Mirror reports that Moyes wants to sign two defenders in January, while he is also keen to bring in a versatile midfielder - and the player who is right near the top of his list of targets is Stoke midfielder Allen.
While Mark Hughes' men have been underwhelming for much of the campaign - though, four points from their last two games has seen them move away from the bottom three - the Welshman has been one of the better performers in the squad, scoring once and providing four assists.
The 27-year-old - who joined Stoke for £13 million in 2016 (via BBC Sport) - demonstrated his goalscoring ability last season, scoring six times in the top flight, and that would be a hugely welcome return if the Hammers did add him to their squad in January.
He may not be the marquee addition that Irons fans feel that they may well need if they are to keep themselves away from trouble, but given the consistency that he has shown during his time in the Premier League with Swansea, Liverpool and Stoke, bringing him to the club may be an extremely shrewd move from Moyes.