On the field, those at Anfield have enjoyed a faultless start to the 2018-19 campaign, collecting maximum points from four fixtures so far.
In contrast, struggles at Old Trafford have delivered just two wins to date, with pressure piled on Mourinho as a result.
The Portuguese has cut a frustrated figure for several months, with a disappointing transfer market doing little to lighten his mood as talk of in-house rifts continues to do the rounds.
Murphy believes Liverpool are showing how things should be done, with the man at the helm leading by example as optimism and belief builds on Merseyside and wanes in Manchester.
The former Reds midfielder told the Daily Mail: “At the moment Klopp is everything that Mourinho isn't.
“Liverpool supporters love his honesty and his enthusiasm. He's ingratiated himself into the club, he's made himself part of it, which is exactly what Mourinho hasn't done.
“Mourinho has been a more successful manager and that's ultimately what counts. But one of the most worrying things about Mourinho is the body language and the lack of fight.
“It's very unlike any other Manchester United team, any Mourinho team to be honest.
“And I don't see how he is going to be able to turn it around that quickly, not unless there's a drastic change in his mentality. I don't see that happening.”
Inconsistency at United could play into Liverpool’s hands, with Klopp’s side looking to force their way into Premier League title contention.
Big-spending over the summer has helped to add greater depth to the ranks at Anfield, with Murphy of the opinion that the Reds are stronger now than they have been for a number of years.
The ex-England international, who spent seven years with his boyhood club during his playing days, added: “This is a good time for us.
“The club have given themselves a really good opportunity by strengthening the squad and creating competition for places.
“I'm not sure they'll beat City; if I'm honest, I think the gap is too big to bridge in one summer – but they'll be challengers at the very least.”