Johnson is currently playing for Wolverhampton Wanderers in the Championship, but Kenny Jackett's side are known to be keen on offloading him.
The central-defender has fallen completely out of contention at Molineux and there appears to be no way that he can make it back into the first-team set-up.
It means that he is likely to be available on a cut-price deal and Sam Allardyce apparently thinks he could be a bargain addition.
Although Johnson is not at the top of the club's wishlist, West Ham are determined not to leave themselves short at centre-back and may make a last-minute move for the 31-year-old.
Impact for West Ham?
Johnson's transfer to the Hammers would be unlikely to go down too well with supporters of the club, considering his loan spell at the Boleyn Ground last year.
Johnson looked well short of the quality required to represent the London outfit during his time with them in the 2013/14 campaign and the club's expectations have increased since then.
West Ham are now challenging for the European spots and the signing of Johnson is unlikely to be one to push them on to the next level.
However, Allardyce will argue that he has proved that he is capable of playing in the Premier League earlier in his career and, as a short-term solution, he could represent a decent backup buy.