Former Roma boss Garcia turned down the chance of moving to Southampton in the summer but is one of the names being considered to take over at the Olympic Stadium.
The clock is ticking for Bilic, who is in his second season in charge but has enjoyed just one Premier League win this term to leave the club in the relegation zone.
Frenchman Garcia, out of the game since January when he was sacked by Roma, is being considered along with former Manchester City boss Roberto Mancini.
Dimitri Payet scored to rescue a point for West Ham against Middlesbrough on Saturday.
The France magician struck in the second-half after Cristhian Stuani had given the visitors the lead.