The Hammers narrowly avoided relegation this season by finishing fourth from bottom in the Premier League, five points above Burnley, and owners David Sullian and David Gold have decided a change in manager was needed.
Current Portsmouth chief Grant was 5/1 with the Harrogate-based firm on Monday but a run of interest forced his odds to be slashed to 1/4, prior to the news of Zola's departure.
Croatia boss Slaven Bilic is next in the betting at 5/1 (cut from 14/1) and Alex McLeish, Martin Jol and Mark Hughes have also been the subject of some interesting market moves.
Alan Curbishley (12/1) and Alan Pardew (16/1) can be backed to make a shock return to Upton Park with former England boss Glenn Hoddle (16/1) and Leicester manager Nigel Pearson (16/1) also near the front of the market.
Sky Bet's Helen Jacob said: "Although Avram Grant remains short priced favourite, there has been money for a large number of other candidates in what is a tremendously lively betting heat today!
"The phone has been buzzing since news of Zola's departure, with some dramatic market moves: Slaven Bilic cut to 5s from 14s, Alex McLeish into 7s from 25/1, Martin Jol 16/1 from 40/1 and Mark Hughes to 8/s from 20s."