However, Pellegrini was adamant post-match he is "not anxious about relegation" from the Premier League but wants to address their poor home form as quickly as possible.
The Mirror are now reporting that the club are lining up replacements in case the team fall to defeat against the Saints on Saturday.
Everton, Newcastle, Southampton, West Ham, Crystal Palace, and even Watford have all been linked to the 30-year-old and on Wednesday, the Nigeria worldwide confirmed he's received offers from England.
Defeat to Southampton could see West Ham drop into the bottom three and the club cannot contemplate the possibility of relegation.
But the Brazilian, who has failed to find the net this season despite starting 15 of West Ham's 16 league appearances, says it is down to him to recapture his form.
Pellegrini was roundly booed by the disgruntled home fans for his decision to replace Felipe Anderson with Sebastian Haller moments after Pierre-Emerick Aubameyang scored the visitors' third. West Ham have spent heavily in signing players since his arrival.
"I think they are always behind the team and, if not, maybe we play here with the stadium empty".
"The Premier League is the best in the world and of course I would love to go back, with Watford or any other team". They then head to Crystal Palace on Boxing Day before seeing out the year with a home match against high-flying Leicester.