The Cherries boss has helped his side into contention for automatic promotion to the top tier this season while playing an attractive brand of football on the South Coast, which has alerted the Hammers.
Despite his experience in the Premier League and proven record, ‘Big Sam’ looks unlikely to be kept on by the London club beyond the summer, with his contract due to expire.
Allardyce has enjoyed a fine campaign with the Irons so far, but his underwhelming 2013/14 season and direct playing style have fractured his relationship with the Upton Park crowd.
Former Manchester United boss David Moyes is also a target, but the Scot is understood to be keen to stay on at Real Sociedad.
Howe has emerged as a genuine alternative, with his antics at Dean Court having garnered much praise.
Just 37, the Bournemouth chief is one of the Football League’s youngest bosses, but has been in management for some time after his playing career ended early due to injury.