British media, including the Daily Mail, Daily Telegraph and Daily Mirror, reported on Monday that 48-year-old Klopp was the favourite for the driver's seat at Anfield Road.
Liverpool bosses fired Rogers after a 1-1 draw with FC Everton on Sunday – although the Mirror reported that it was actually Klopp's expression of interest that triggered the move.
"The search for a new manager is underway," was the only indication in a joint statement from the Liverpool FC owners on Sunday.
A decision is expected before Liverpool's next game in the Premier League against Tottenham on October 17th.
Jumping across the Channel and into the Premier League would be a surprise move for former BVB manager Klopp, who had said after stepping down earlier this year that he would take a year off from football.
Taking on the Liverpool job would make him the only German currently managing a Premier League team.
He'd be hoping to do better than the last of his countrymen to manage one of England's elite squads, Felix Magath, the former Mannschaft player and manager at Stuttgart, Bayern Munich and Wolfsburg who took over at Fulham in February 2014.
Magath was fired after just seven months following a run of eleven league games without a win.
But English commentators were confident Klopp would be far from that disastrous performance.
"It has to be Jürgen Klopp. He's the perfect man to take over from Brendan Rogers. He has the qualifications and, more important, he can give the fans back their belief in the team," wrote Jan Molby in the Liverpool Echo.
"Klopp has an aura," former England midfielder Jamie Redknapp wrote in the Daily Mail. "He's a bit mad but I like him and he'd be box office for the Premier League.
"There are good players in the squad and if someone comes in and really lifts them, there's every chance they could get [into the top four] again this season."
Klopp is famed for his passion and enthusiasm. His one liners are legendary, as are his celebrations.
But English journalists might want to take his heart-on-the-sleeve style into consideration before they meet him. In 2014 he stormed out of an interview after a journalist suggested Dortmund's Champions League campaign was over following a 3-0 loss to Real Madrid.
On Twitter, commentators from England and Germany were excited by rumours of Klopp's arrival.
"I would find that so cool!" wrote one German BVB fan.
Referring to one of Klopp's famous caps with the word "Pöhler" (Ruhr region slang for street footballer), another fan suggested Klopp could combine an adopted club with his own language: