However, Liverpool a have also made contact with former Real Madrid and AC Milan boss Carlo Ancelotti.
Klopp is understood to be the number one choice of the two and talks are set to resume later this week.
Liverpool are searching for a new manager after sacking Brendan Rodgers after the Merseyside Derby on Sunday, that ended in a 1-1 draw.
The German left former club Borussia Dortmund after an amazing end to last season, which saw them rise from the relegation zone to qualifying for the Europa League
Klopp spent eight year at Dortmund and in that time took them from a mid-table side and turning them into Germany's second bests team, winning the Bundesliga title once and reaching a Champions League Final.
Sky Sports' Vinny O'Connor said "One of the reasons these two managers are of interest is that they are both out of work at the moment, which makes a deal relatively easy to do.
"But with Klopp, he has a track record of working with younger players, improving them, but also losing his best players and still being able to produce a successful team.
"His record in the Bundesliga shows that and he also got Borussia Dortmund to the final of the Champions League, and if you look at the way Liverpool have operated in recent years you can see why he's their number one choice."
Source: DSG