Van Dijk is braced for a barracking from the home crowd at St Mary's on Sunday following his switch to Anfield last month, which followed on from a summer of acrimony in which he had been blocked from leaving.
Pellegrino, though, wants everyone to move on from the protracted saga, with Southampton having made a healthy return from the £11.5m they paid to bring Van Dijk from Celtic in September 2015. Van Dijk stayed with the Saints until the transfer window opened again this year although signalled his intentions to leave the south coast club.
'I think how football fans are, they want to disturb everything we try to do to help their team and they will whistle.
"It is not the most lucky season and of course there is a kind of frustration in him", said the Reds boss.
Klopp said: 'We are all human beings, we are influenced by circumstances.
"I don't think it is a big thing at the moment for most of the players but with the long term injury how can you not think about it?" We have to go there and play football and nothing else. Is it nice? I don't think so.
"Every single person has an opinion about everybody, and we have to respect that, but I think our fans are really respectful with ex-players". I don't think anyone at Southampton will say we've stolen him or something.
"We understand it was not the challenge before the situation but it was another challenge before that".
Liverpool have signed a number of players from Southampton in recent years and Klopp is surprised by the number of top-flight players to have graduated from the Saints.
However, in his latest outing, Van Dijk was involved in a controversial last-minute penalty collision with Erik Lamela that cost the Reds two points as Harry Kane scored a penalty to rescue Spurs a point against their top-four rivals in a thrilling 2-2 stalemate. 'Adam is not happy about the reaction as I was not happy about it.
"Alex Oxlade-Chamberlain is obviously from there, he took a different way, and Theo Walcott as well". Theo Walcott as well, a few good players from there - maybe we should have a scouting department in Southampton! He did well in our club and he left some profit to the club and we have to grateful. But it just shows the good work they do there.