As recently as Wednesday, BetVictor had not included the Hammers amongst the genuine contenders to land the 34-year-old, with Sunderland listed as 4/1 favourites while Hull City were second at 6/1.
Everton's Phil Jagielka
However, West Ham have swiftly moved into the running, according to the bookmaker anyway, and are now placed in between the previous two favourites at 5/1 (via BetVictor).
It would perhaps not surprise supporters if the Irons did try and target the England international this month given that they have been linked with a move for another one of the Premier League's top defenders.
Southampton's Jose Fonte has been the subject of a bid
The Sun reported on Wednesday that West Ham had made a £4 million move for Southampton's Jose Fonte, so it perhaps makes some sense for the club to have alternative targets.
And given that Jagielka and Fonte are of similar ages and both boast plenty of top flight experience, it is clear to see why the bookmaker are suggesting that West Ham are contenders to secure the Toffees man's signature before the current window closes.