The Argentine manager came to England after a fairly successful run with Spain's Alaves, but Southampton sit 17th in the EPL, just one point above the relegation zone with eight matches to play, reports Efe.
Saints are now one place and one point above the relegation zone and realistically need three more wins (they only managed five in 30 games this season under Pellegrino) to stay out of the bottom three.
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Other nations, like Japan, South Korea and Brazil have also called for similar treatment. Sure, they won't stop driving, but they will have less money to spend on other stuff".
Southampton have already spoken with Mark Hughes as he has looks firm favourite to take charge of the club.
"I have to say thanks to Ralph (Krueger, club chairman), to the board, to Les (Reed, executive director) and Ross (Wilson, director of recruitment and scouting), all my technical staff and all the Southampton employees".
Frenchman Claude Puel was sacked after leading the team to eighth place and the League Cup final last term, and has since thrived after taking over at Leicester earlier this season.
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I am convinced that is a good basis for the team on the spirit front and overall I am very happy with the spirit of the players. The win leaves the Gunners sixth, 12 points behind fourth-placed Liverpool , who are in the final Champions League spot.
They now have only one win in seven and have conceded 11 goals in the last three matches, plus there is total civil war breaking out between fans and club.
Of this season's contrasting fortunes, Pellegrino said: "We have been living a hard season for many reasons".
The struggling south coast side, 17th, are now keen to appoint a manager with Premier League experience who can come in ahead of Sunday's FA Cup quarter-final clash away to League One side Wigan.
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Like Hughes, Silva was sacked by a Premier League club in January - in his case Watford - and he has also been linked with a move back home to Portugal to take over at Benfica next season.