Pellegrino took over at St Mary's on a three-year deal last June, but Southampton announced in a statement that they had parted company with the 46-year-old with the club only one place and one point above the relegation zone.
Benfica are planning a run at former Watford head coach Silva should Rui Vitoria fail to win the Portuguese title.
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.
The club sit 17th in the league on 28 points, a point ahead of Crystal Palace and Stoke, with eight games left to play.
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"From the beginning I tried to do my best every single day, I tried to give back to the club and tried to represent the club with pride", the letter began.
"Some moments we managed this, some moments we couldn't, but we have been living a hard season for many reasons".
They were beaten 3-0 by Newcastle on Saturday and their next league game is away to West Ham United on 31 March.
Southampton legend Matt Le Tissier believes Mauricio Pellegrino was sacked because of his negative style of football.
As reported by The Scotsman, Celtic manager Brendan Rodgers' achievements in his first two seasons in Glasgow have lead to some strong interest from Premier League club Southampton.