The 29-year-old moved to Anfield from Aston Villa in July 2011 and has made 91 appearances, scoring seven goals, for the Reds.
The England international has struggled to cement a place in the first team, particularly in his favoured forward position, and Liverpool boss Brendan Rodgers has told the former Middlesbrough man he is free to leave.
West Ham boss Sam Allardyce will be hoping Downing can provide the crosses to help former Liverpool striker Andy Carroll thrive at Upton Park.
Alan Pardew said last week he was "very interested" in the player, but it seems Joe Kinnear has other ideas.