The 31-year-old defender, who has been released by Sunderland, is understood to be training with the Hammers in a bid to maintain his fitness.
Bramble's contract at Sunderland expired at midnight with the player having been told he would not be offered an extension.
He made 51 appearances for the Black Cats after being signed for £1million from Wigan by then manager Steve Bruce during the summer of 2010.
Bramble, who began his career at home-town club Ipswich, completed a big money move to Newcastle during the summer of 2002, but struggled to fulfil the potential in which then manager Sir Bobby Robson had invested so heavily.
He left St James' Park for free after five years and joined Wigan, where he rebuilt his reputation under Bruce.
He had been training at Ipswich and Turkish club Besiktas have been checking him out in recent weeks - a deal that would interest him (apparently).