Marco Silva, who was reported to be an early contender for the post, is not though to be under consideration - and, in any event, the former Watford boss is being courted by Benfica.
Bilic managed the Hammers in London for two years, from 2015 to 2017, before getting sacked in November 2017 after a 4 - 1 defeat by Liverpool.
The 54-year-old, who made just over 50 appearances for Saints as a player, is the leading candidate to take over at St Mary's following the sacking of Mauricio Pellegrino on Monday night.
Hovering just one spot above the relegation zone Southampton are in danger of losing the top-flight status they acquired six seasons ago.
Mark Hughes is in talks to become Southampton's next manager, FP understands.More news: Jose Mourinho told to make these two Man United signings this summer
More news: Man Utd midfielder Lingard: Rashford always ready for starting call
More news: Killed As Private Jet From UAE To Turkey Crashes In Iran
Current Southampton assistant coach and former goalkeeper Kelvin Davis took training on Tuesday after Pellegrino's departure. The Portuguese coach is available, having lost his job back in January, but will no doubt be conscious of a second successive Premier League relegation, having suffered the dreaded drop with Hull last season.
The struggling south coast side, 17th, are now keen to appoint a manager with Premier League experience who can come in ahead of Sunday's FA Cup quarter-final clash away to League One side Wigan.
However before that Hughes had several short-term stints firstly in charge of the newly-funded Manchester City before moving on to both Fulham and QPR.
The club sit 17th in the league on 28 points, a point ahead of Crystal Palace and Stoke, with eight games left to play.
"Sparky" will be expected to make his return to Southampton as a manager in the coming days, after spending two years from 1998 to 2000 with the Saints as a player.