West Ham are in need of a back-up to Robert Green, with Jan Lastuvka having returned to Shakhtar Donetsk following his loan spell.
Southampton's financial problems have meant that Davis, their 32-year-old captain, is available on a free transfer.
Davis has experienced Premier League football before, having played for Sunderland during their 2005-06 campaign, which ended in relegation.
Meanwhile, Lucas Neill will consider offers from other clubs according to his agent, Paddy Dominguez.
West Ham had accepted that Neill will leave having rejected a new contract offer, but the captain is still believed to be keen to remain in London.
Neill has been closely linked with a move to team up with Harry Kewell at Galatasaray and although that move has not yet come to fruition, a likely shift from Upton Park still seems likely.
'We will wait one week for West Ham to get back to us and if we hear nothing we will have to open dialogue with the interested parties from Britain and the continent,' said Dominguez.
'Lucas would love to stay with the club and his young family is settled in London.
'But he has perhaps two top-class seasons left in his career and he certainly wants to know where he will be playing next season.
'Pre-season training will start soon and that is why we want an answer from West Ham as soon as possible. This is a big factor.
'Galatasaray are interested and they have been since last January but this is not about money.
'Lucas is open minded about the whole thing. If he eventually leaves the Hammers he would want to sign for a club that will challenge for silverware and be in a position to play in the Champions League.
'Lucas has been in the game for a long time and he would like to finish with a trophy or two.'