There were plenty of Saints fans underwhelmed by Puel's solitary season in charge, despite reaching the EFL Cup final and securing a top-half finish, so plenty will be expected of Pellegrino who took Alavés to the Copa del Rey final and a ninth-place finish in La Liga last term.
We take a look at the top five Saints transfer stories according to the Football Whispers Index, which scores the validity of transfer rumours out of five, to see who could join move to St Mary's this summer.
Gregoire Defrel – Sassuolo – 1.7
Despite finishing eighth last season Southampton only scored 41 times. Only five teams found the back of the net less frequently.
The 26-year-old Frenchman scored 12 times in just 23 Serie A starts last season while also netting twice in four Europa League run-outs for the Neroverdi.
The Saints will have to act quickly if they are to sign the £18million-rated forward.
Ridgeciano Haps – AZ Alkmaar – 1.7
Another area which could need attention this summer is left-back. Current No.3 Ryan Bertrand is a Chelsea transfer target after two impressive seasons at St Mary's following a move from Stamford Bridge in 2015.
Haps made 33 Eredivisie appearances for AZ last season, suppling two assists, while also featuring eight times in their Europa League run. He averaged an impressive 2.9 tackles per game as well as 2.6 interceptions and 3.1 clearances.
Bacary Sagna – Manchester City – 1.5
In truth, someone to rival Cédric for the No.2 shirt would have been on the wish-list of whoever replaced Puel. But the fact the Portuguese is a £15million target for the Bianconeri will only increase the need for depth.
The former Arsenal defender has spent the last ten years in England following a move from Auxerre in 2007 and has been a model of consistency.
Although at 34 the French international would not be a long-term option he would provide stability in the short term and bring a wealth of experience to St Mary's.
Riza Durmisi – Real Betis – 1.4
Normally when Liverpool and Southampton are mentioned in relation to the same player it's because the Reds are trying to sign a Saint.
Last season Durmisi averaged 1.7 tackles per game, as well as 1.1 interceptions and 1.7 clearances. He scored once and supplied four assists from left-back in La Liga.
Jordan Amavi – Aston Villa – 1.3
The French Under-21 international impressed when he moved to Villa Park in 2015 but an ACL injury ended his first season in England prematurely.
He only featured 17 times in the Championship last season and Villa turned down a £25million bid from an unnamed Chinese club for his services in January.
Now Amavi has stated his desire to leave the Midlands and he is, according to reports, available for transfer.