Southampton haven't had a great season but they have picked up home wins over other struggling sides of late and will understand the importance of doing so again here.
The club's hierarchy have been considering Pellegrini's position for several weeks and were minded to sack the Chilean in the event of defeat on Saturday, but a dominant display has eased the pressure on the manager, whose side moved four points above the relegation zone.
For Pellegrini, defeat at St. Mary's against the fighting saints is not an option, as the club is already thinking of replacing him with free-riders like David Moyes and Chris Hughton, or possibly coaching first-team coach Enzo Maresca to janitor at the club and do not follow Pellegrini when the manager leaves.
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This is not the only alleged point of contention in the West Ham dressing room, with the Sun also claiming that Pellegrini's stubborn tactical outlook and the performance of director of football Mario Husillos have also acted as lightning rods for criticism.
"That's not a problem of the job - the problem is to try to improve what you're doing because we don't have the results".
At his press conference before Thursday's game in Rush Green, the manager was immediately asked if he would be working Monday morning once the television cameras were off. To be fair I have been here five years now and we always have these blips, but the boys come out of it and hopefully this is the turn where we can start moving forward and getting ourselves up the league. "After that you can work on tactical, physical and technical work, but the performance of players decide the games".