Liverpool trail league leaders Manchester City by a point going into Sunday's final fixtures.
Victory over Wolverhampton Wanderers in their final match at Anfield will put them on 97 points, yet that will only be good enough for second if City win at Brighton and Hove Albion.
While the season could end without silverware, Grobbelaar, who won six top-flight titles with Liverpool, believes the club are heading in the right direction under Klopp, seeing similarities between the German and the club's most famous boss.
Shankly spearheaded Liverpool's rise to the top of English football in the 1960s, winning three First Division titles and establishing the foundations that would help the club dominate the domestic game for almost three decades.
"From (Shankly) until Juergen Klopp I don’t think there’s anyone that can be talked about in the same breath as those two," said Grobbelaar, who played for Liverpool for 13 years from 1981.
"Because Shankly was an icon. He knew the fans, he got the fans, he’s charismatic. Now we’ve got exactly the same.
"(Klopp) has got the charisma to take this team much further than it is today."
Grobbelaar had a reputation for flamboyance, risk taking and spectacular shot-stopping in his time at Liverpool, qualities that some have said belong to current keeper, Brazilian Alisson Becker.
While Grobbelaar had words of praise for Liverpool's whole defence, he said Alisson was a special talent.
"(He is) one of the best goalkeepers in the world," said Grobbelaar.
"One of the reasons why we are so close in getting the Premier League."
Grobbelaar, who also won the European Cup for Liverpool in 1984, said the current side are not far from building a squad capable of clinching their first league title since 1990.
"We are not that far away from producing a squad that’s going to surpass Manchester City," said Grobbelaar, who was in New York to promote Liverpool's summer series of friendlies at Yankee Stadium, Fenway Park and Notre Dame.
Grobbelaar said the sport was growing in the United States, where baseball, American football and basketball have long dominated.
"The numbers are slowly getting there. With the amount of young people playing the game of soccer as opposed to American football, it’s growing all the time," said Grobbelaar, who had his "baptism in professional football" playing in North America for the Vancouver Whitecaps.
"When I moved from Vancouver Whitecaps to Liverpool I didn’t think that I would ever come back to the United States for football," Grobbelaar said. "I never knew how big Liverpool could actually have got (in the U.S.)."
(Reporting By Amy Tennery; Editing by Toby Davis)
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