Joey O’Brien is having to cover the position temporarily for the Hammers, with first-choice George McCartney sidelined by a knee injury.
The East London club are thought to be keen to bring in a replacement during January and see Aston Villa’s out-of-favour full-back Warnock as the ideal candidate.
The former England international has just returned to the club following a loan spell with Championship outfit Bolton, who are thought to be keen to make the move permanent.
However, Allardyce is confident that he can persuade the 31-year-old to choose Upton Park over the Reebok.
The Villains are unlikely to demand a high figure for his services, which will further tempt Allardyce, who confessed that signing a left-back is a priority:
“Yes it would be. One of my priorities would be a left back if there is one possible for us to bring in.”
Warnock left boyhood club Liverpool to join Allardyce at Blackburn in 2007, where he enjoyed a successful spell.
After two years at Ewood Park he moved on to Aston Villa, but has fallen out of favour in recent years.
Warnock has made two senior appearances for England, and was part of Fabio Capello’s 2010 World Cup squad.