The club face Watford when they return from the worldwide break which gives Sullivan and co-chairman David Gold some time to consider their decision.
Beleaguered West Ham boss Slaven Bilic will know that defeat on Sunday could see his time in charge of the Irons come to an end, with patience at boardroom level running thin. "As players we can be professional and do our best".
Two minutes later it was 2-0 as Salah's corner nearly forced Mark Noble into an own goal and when Hart pushed the ball out, Matip was on hand to prod in the rebound.
With 12 goals in 17 games, Salah has certainly settled into life at Liverpool; factor in his four assists and his contribution to Klopp's side can not be underestimated.
Salah stayed rather too high up the pitch, with he and Firmino playing into the hands of West Ham's three-man defence.More news: Dwyane Wade sees some of 2014 Miami Heat in current Cleveland Cavaliers
Just as it looked the home side might mount some form of comeback, Liverpool struck again through Alex Oxlade-Chamberlain netting his first Premier League goal for the club.
Klopp revealed how Liverpool had hardly practised the 4-4-2 formation that they deployed against West Ham. Of course we're going to discuss this defeat because it's not the first one. Meanwhile, West Ham remain a point above the relegation zone after their winless run was extended to four games.
"I watched him a lot at Chelsea, and it didn't really take off for him when he was a Chelsea player", he said on BT Sport.
A mis-judged timing on the path of Joe Gomez saw Manuel Lanzini chest down Andre Ayew's cross to the back post before the Argentine poked the ball past Simon Mignolet to bring the Hammers back into the game but their celebration was shortlived.
Liverpool's two-goal lead was restored as Oxlade-Chamberlain squeezed a shot past Hart, after the keeper kept out his initial effort.
Asked if he still felt he had the backing of the club's hierarchy, Bilic replied: "It's hard for me to talk about that now".