Southampton parted company with Claude Puel following the Frenchman's first season at St Mary's.
The Saints are looking for their new boss following Claude Puel's recent departure after just one season at St Mary's, and The Sun suggests that they have already earmarked the man they want to replace the Frenchman.
According to The Sun, the Southampton hierarchy hope to reach an agreement with the Argentinian - who is also being considered by Crystal Palace as they look to replace Sam Allardyce - by the end of the week.
If Pellegrino is interested in a switch to the south coast, it may perhaps be fair to suggest that it will not be difficult for the two parties to agree a deal following his departure from Alaves after this past season.
Claude Puel recently left Southampton
And Southampton fans will be hoping that his record with the club - if he is appointed - ends up looking rather familiar to the one that Mauricio Pochettino boasted with the side following his arrival from Spain in 2013, when he helped take the side to another level following Nigel Adkins' incredibly impressive spell and highly controversial departure.
Though Puel was criticised throughout his spell, and rarely had the full backing of the fans, he does leave his successor in a good position, having led the side to eighth in the Premier League while also competing in other competitions, and the Saints supporters will be hoping that Pellegrino, if he is the man appointed, can lead the side to that next step.