The Liverpool Echo reported that talks had already begun with Klopp on Monday – just 24 hours after former manager Brendan Rodgers was removed – and that the German was keen to join.
Liverpool bosses hope to have the ink dry on Klopp's contract by the end of the week, the paper reported.
Klopp has always been the favourite to take over at Liverpool, although there was a suggestion that former Chelsea manager Carlo Ancelotti might also be in the running on Monday.
In fact, the Echo reports, bosses at Anfield have had their eye on Klopp since 2012, although he said at the time that he wouldn't leave Dortmund and the Bundesliga for the Premier League.
But following his resignation from the Rhine-Ruhr team at the end of last season, he appears to be itching for a new challenge – despite rumours that he would take a year off football.
"Jürgen Klopp, who lives out his passion and love for the game, would fit wonderfully to Liverpool's mentality. He would be the ideal solution for me," said former Liverpool midfielder Dietmar Hamann on German Sky Sports on Monday.
"The emotions in Liverpool need to be awakened. The people have lost faith in the team and it is important that someone comes who rouses the people again and gets things going."
Meanwhile, Mannschaft and Liverpool midfielder Emre Can told Die Welt that "Jürgen Klopp is certainly a great coach. I think he has demonstrated in Dortmund that he can work at any team in the world".
Klopp's record at Dortmund includes Bundesliga titles in both 2011 and 2012, the year they won the domestic double after routing Bayern Munich 5-2 in the German Cup final when Robert Lewandowski scored a hat-trick against his future employers.
He also guided Borussia to the 2013 Champions League final, when they lost 2-1 to Bayern.
The charismatic Klopp speaks good English and said last year he wanted to work in the Premier League.
"I think that apart from Germany, England is the only other country where I should work," Klopp told German magazine Focus last year.