The German has just delivered the club’s sixth European Cup, their first major trophy for seven years, and guided them to a second-place finish in the Premier League with a record runners-up points tally of 97.
Klopp’s current deal expires in 2022 but FSG, essentially principal owner John W Henry, chairman Tom Werner and FSG president Mike Gordon, want to secure him for longer, Press Association Sport understands.
“When the players threw Jurgen in the air last night after our victory, they were acknowledging what all Liverpool fans know- he is beloved by all who appreciate football,” said Werner.
“Jurgen is a brilliant coach but just as importantly he is a humble and caring man who is happy today because he knows what yesterday meant to our supporters.”
For his part Klopp was amazed by the 500,000-plus turn-out for the team’s victory parade in the city on Sunday.
“I don’t know exactly how many people live in Liverpool but it does not look like there is a lot of space for fans of other clubs,” he told LFC TV.
“It is incredible. If you are a young kid, which do you support?
“There are not a lot of opportunities or options other than Liverpool. You see in the eyes how much it means. It is really intense.”