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.
Saturday's Champions League final victory over Tottenham was not a classic but nobody in the red half of Merseyside cared as the club's faith in Klopp was finally rewarded with his first piece of silverware after four seasons at Anfield.
His 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, reports claim.
"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."