The Reds are currently top of the table by two points ahead of the reigning champions, having played one game more, with just over a month of the 2018-19 campaign remaining.
A 2-0 victory over Chelsea at Anfield on Sunday saw Klopp's men regain the initiative in the title race and a first domestic crown in 29 years could be on the cards if they win their remaining four matches.
Since joining the club in 2015, Klopp has failed to win a single trophy, despite reaching three major finals.
However, he enjoys hero status among the club's supporters for his fierce displays of passion on the sidelines and for his preferred style of play which has produced so many memorable performances.
Lawrenson, who won five league titles with Liverpool between 1981 and 1988, has lauded the German coach for his unique managerial qualities, insisting he deserves plaudits even if his side fall short in their latest bid for silverware.
"Finishing second would be tough to take, but it wouldn’t undermine Klopp at all," he told Paddy Power. "He’s seriously improved every player at the club, which is a feat in itself.
"I remember, when he came to Liverpool, saying that a lot of the other teams in the league would be thinking they wished they’d got him. What you see is what you get with Jurgen.
"He gets the football club, he gets the history. Quite a few of the managers in the past have tried to play the history down, as if it weighed a little too heavy on their shoulders, and didn’t like having the former players around.
"But Klopp gets it. Every time you go to the training ground, he laughs and says ‘here they are again, the winners’. He always stops for a chat and picks your brain about the club. He’s embraced it.
"So his job has been about more than winning this title. Though it’d be nice for him to win it, because every time he’s asked about not winning something with Liverpool, I’m sure he’s wanted to chin whoever’s asked it.”
Next up for the Reds is a Champions League quarter-final second leg against Porto on Wednesday, which they will be expected to win after a 2-0 victory in the reverse fixture last week.
A semi-final berth against either Barcelona or Manchester United beckons and Lawrenson hopes his old club face the former so that the likes of Lionel Messi and Luis Suarez can arrive for another famous night at Anfield.
“I think Liverpool will get past Porto," he added. "I was at the home game, and Porto looked average, they couldn’t cope with the intensity of a top team.
"Then you have to believe it would be Barcelona in the semis. That would be just wow. Having played in many European nights, they’re always fabulous, but that would be way, way up there, the game at Anfield.
"Just the fact that it’s Lionel Messi, the greatest player ever born. But also Luis Suarez coming back. That tie would be 50-50, but Klopp has an unbelievable record over two legs with Liverpool."