Slaven Bilic's side found themselves 2-0 down after just a matter of minutes and although they laid siege to the Foxes goal in the second half, could not find an equaliser.
Snodgrass replaced Dimitri Payet in January and while nobody was expecting the Scotland international to be anywhere near the Frenchman's level - even though he cost the same amount - fans are already growing impatient with his performances.
Snodgrass has been full of big talk since arriving at the club but has failed to back it up so far.
A distinct lack of pace and no sign of a goal or assist so far, Snodgrass has been hauled off or started sub in most of the games so far after failing to impress.
That was the case again last time out when he was introduced as a second half sub and yet again failed to impress.
Hammers legend and co-commentator on the match Tony Gale tore into the playmaker after his worst display yet in the claret and blue.
Gale, who played 300 games for the East Londoners during a decade at the club, could not hide his disappointment at the former Leeds United man's performance as he made a number of observations on the Scotsman's woeful contribution.
West Ham United's Robert Snodgrass reacts
"Dear oh dear, Snodgrass keeps passing to (Sam) Byram when it's really his responsibility to be getting the crosses in and making things happen from that side," Gale said.
"He's got the ability but he keeps passing the buck and he's running Byram ragged out there.
"He's been really disappointing I must say."
One of the main reasons Bilic signed Snodgrass was because he would not need time to settle into the Premier League.
But he has looked like a fish out of water for most of his time in East London so far and there are already serious question marks over his future.