The Reds youngster is set to become a free agent in the summer when the 23-year-old’s current contract expires at the end of the season.
Flanagan impressed as he started his first Premier League game since his 18-month injury lay-off at Anfield on Wednesday night.
The England international earned plenty of plaudits after he tamed Manchester City forward Raheem Sterling in Liverpool’s 3-0 win at Anfield.
Former Liverpool striker Mellor expressed his disbelief that the Merseyside outfit haven’t offered Flanagan a new long-term contract.
“How this lad hasn’t had a new deal sorted out yet I don’t know,” Mellor told Liverpool Echo.
“We always seem to admire the continental players and get them signed up swiftly but here we have a lad who is full of passion and desperate for a long term future at the club he has supported all his life.
“Jon’s showed he has quality in a specialist position. He might not be the quickest or most creative but you need defenders to defend and he does that so well.
“We’re missing that local input since Jamie Carragher and Steven Gerrard both departed and Flanno is the lad who can carry that torch forward I feel.
“I hope the Reds just get a new contract with him sorted out soon.”
Flanagan will be hoping to retain his place in Jurgen Klopp’s starting line-up for Sunday’s trip to Crystal Palace.
The Reds are in ninth place in the Premier League table and nine points behind fourth-placed Manchester City.
The 18-time English champions face Manchester United in the first leg of their Europa League last-16 clash on Thursday night.
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