Gerrard has been widely criticised for his performances in recent weeks and was left on the bench for Liverpool's 1-0 victory over Stoke City last Saturday, a result that ended a run of five games without a win in all competitions.
With that game marking the 16th anniversary of Gerrard's first-team debut for Liverpool, reports emerged that the 34-year-old was unhappy with Rodgers for omitting him.
Those claims were quickly dismissed by the former England skipper, who took to Instagram to label reports of a bust-up between himself and the manager "nonsense".
And, speaking ahead of Tuesday's Premier League trip to Leicester City, Rodgers revealed that Gerrard - whose current deal expires at the end of this season - has been given the chance to extend his stay at Liverpool.
"Firstly in terms of the club, the latest is that Steven Gerrard and his representatives have been offered a contract here and at this moment in time that's where it's at," he said.
"For Steven, everything he's done in the period of time he's been here, for 16 years, is a man who deserves utmost respect and time to consider that. It's not just a big move in his life in terms of commitment [and he] needs time and respect to look at that.
"It certainly won't be money. I've spoken to him enough times at length and that won't be the case. Between now and the end of the season he deserves the opportunity to think."
When asked whether there had been a rift between himself and Gerrard, Rodgers added: "No. 150 per cent no."
Liverpool remain five points off the top four following the Stoke victory, which Rodgers is eager to build on against a Leicester side that is without a win in eight games and sits bottom of the Premier League.
"They've [Leicester] got good players, players who are a threat. I looked at [Jamie] Vardy when I was at Swansea, when he was at Fleetwood - he's very fast and dynamic and can get goals," Rodgers added.
"We arrive there having to be fully concentrated and just want this momentum to keep going."
