Liverpool sacked Kenny Dalglish on Wednesday after finishing a disappointing eighth in the Barclays Premier League – despite winning the Carling Cup and being runners-up in the FA Cup.
Fenway Sports Group, the Merseyside club's owners, have had an approach to speak to Wigan manager Roberto Martinez accepted and were also keen to interview Rodgers.
But it is understood the 39-year-old has rejected the approach.
Meanwhile, Wigan chairman Dave Whelan has paved the way for Martinez to become the next Liverpool boss.
Long-serving Whelan gave the go-ahead for talks to take place soon between the former Motherwell player and the Anfield club.
He said: "If Liverpool are serious then I may lose him."
Whelan has always said he felt the Spaniard was destined to manage one of Europe's top clubs and he would not stand in his way should one come calling.
The Wigan chairman revealed the Reds made an approach yesterday and he gave Martinez permission to speak with them.
"When they phoned me I was actually with Roberto having a meeting on what we are doing next season," he said.
"I did promise Roberto and I have always said when a big club comes he would have permission to talk to them and they don't come any bigger than Liverpool.
"I gave him permission and he will be talking to them soon– I don't know when, but quite soon.
"When Liverpool sacked Kenny I have to say I thought they would be knocking on the door and sure enough they are. I would love to keep him but I have told Roberto he has my permission to speak with Liverpool."
Meanwhile, John Barnes has backed axed Chelsea boss Andre Villas-Boas to be the next Liverpool boss.
"Villas-Boas is a good manager," Barnes said. "His footballing philosophy and his style of play fit. He proved he could win things at Porto.
"He was at the right club at the wrong time with Chelsea.
"There were a lot of problems for him there because of the older players. He is a proven manager and he would be coming in to a humbler squad than the one at Chelsea."