The Hammers have endured a difficult start to the new season, winning just one of their six games so far.
The likes of Rafa Benitez and Roberto Mancini had been linked with the Hammers hotseat.
But a win over Huddersfield and draw at West Bromwich Albion had briefly lifted the gloom gathering over the London Stadium.
Defeat to bitter rivals Tottenham Hotspur, and some questionable tactical decisions from Bilic have once again cast doubt over the Croatian's future in East London.
Reports in the likes of The Sun claim the Hammers made an approach for former Borussia Dortmund manager Tuchel.
The 44-year-old left Dortmund in the summer and The Sun reports that he turned down West Ham's advances.
But West Ham supporters website Claret & Hugh, which claims to have an well-placed source at the club, says the owners have not approached Tuchel and still back Bilic.
However, that has cut no ice with bookmakers who have installed Bilic as the new 7/4 favourite to be sacked, replacing Everton boss Ronald Koeman at the top of the list, as highlighted by Oddschecker.