The reigning Premier League champions returned to winning ways away at Sheffield United on Sunday to move to within one point of fifth placed Chelsea, who they face on Thursday, and two points off fourth placed West Ham.
The Hammers lost at Manchester City on Saturday and face Leeds United in their next game, followed by clashes with Manchester United and Arsenal.
Many feel that the Hammers will drop out of the top four, but Gillespie is of the view that should not be taken for granted, even though he thinks Liverpool will finish in the top four.
“I think everybody will expect West Ham to slip out; that is not to say they will do that”, Gillespie said post match on LFC TV.
“If you look back to when Everton were in a similar position under David Moyes and everyone thought they would drop out and they didn’t do that.
“I firmly believe that Liverpool will be in the top four.
“But take nothing away from West Ham. They have had a fantastic season.”
Liverpool have beaten West Ham in both of the league games between the two teams this season, but the Hammers still hold a two-point lead over the Reds in the Premier League.
