Klopp has branded Mignolet as one of the most intelligent goalkeepers he's worked with since his arrival but the Belgian has come under fire for yet another error during the 2-2 draw with West Bromwich Albion on Sunday, meaning the Reds surrendered a lead they had taken shortly beforehand.
Molby wrote in his Liverpool Echo column: "As for handing Simon Mignolet a new contract, there's a lot of good will around the decisions that Jurgen makes and many people believe if this is what the manager feels then they will support that.
"The goalkeeper knows himself if he comes and doesn't quite get there and ends up in the back of the net, he'll be in for criticism. That's part of the job that he has.
"I do believe he's come on a lot, mentally he's a lot stronger, and he'll just get on with it."
Mignolet has had patches of good form, but the winter months don't seem to treat him kindly as this time last year he was dropped by Brendan Rodgers with Brad Jones given a short spell in the side.
There have been times when he has been quite solid, and his shot stopping isn't in question but there are times when he just provides uncertainty to the back line, and then there is the odd howler chucked in for good measure which means he is firmly out of favour with fans.
The Belgian is capable of putting some good form together but there are too many occasions when he has made errors that have cost the sides points, and he needs to address it. He has come back from poor form before, but he needs it to become a distant past rather than wait another year for another bad patch.
Klopp is keen to give all members of his squad a go, and in a way it is good management in terms of putting his faith in Mignolet, now he needs to work his magic with him.