Molby thinks not winning a trophy worked against Klopp's predecessor.
The Reds' busy schedule continues when they travel to Carrow Road to take on Norwich City in tomorrow's lunchtime kick-off in the Premier League before they host Stoke City in the second leg of their League Cup semi-final.
In his Liverpool Echo column, the Reds legend wrote about the credit he would then have behind him in order to build the side in his image, and lay the foundations for success.
"We have a big League Cup semi-final which could take up energy so I think it's going to be hard, and it will make Norwich even trickier. Our inconsistency means you just don't know what to expect, Jurgen will be aware of this but he knows he can't change it overnight," Molby wrote.
"If he has the choice, he would rather see them produce the top form in the Stoke game, and the fans will feel the same.
"Just ask Brendan Rodgers how important winning a trophy for Liverpool is.
"When it started to go wrong, people said how he hadn't won a trophy but came close on a few occasions. Then you have to ask if he's a manager who wins trophies, or was he a nearly man?
"Klopp is a man who wins and he's proved it in the past so he's keen to put a trophy on the table. It is his priority. They are two matches from it, so go for it."
Given the position in the Premier League seems to be an uphill battle for the side to achieve their objective of a place in next season's Champions League, many would be of the opinion that the clash with Stoke City on Tuesday is far bigger as the Reds look to take another step in ending their four-year wait for silverware.
Molby is right that when the topic of Brendan Rodgers' future came up, his lack of silverware worked against him when it came to making the case for and against his dismissal.
Klopp can demonstrate that FSG's trust and decision making was not misplaced if the German wins a trophy at the first available opportunity.
Firstly, the Reds would like to make their league position a little more healthier with victory over the Canaries in the League tomorrow.