Rodgers confirmed the offer at a news conference ahead of Tuesday's Premier League match at bottom of the table Leicester City.
"He has been offered a contract here," Rodgers said, adding that Gerrard was now considering it.
"He has been here for 16 years and he is a man who deserves the utmost respect and should be given the time to consider it.
"It is a big move. The one thing it won't be about is money. I have spoken to him enough times and at length, and that won't be the case.
"One thing is clear, I love working with Steven Gerrard. He is arguably the best player that's ever played in the Premier League and I have enjoyed every single minute of working with him here and I hope that continues.
"Now he is at the stage of his career where it is not so much the number of games he plays now, its about the level of the game and I will work with him and look at it."
Former England skipper Gerrard, who has played nearly 700 games for Liverpool since making his debut 16 years ago last weekend, came on as a late substitute in Liverpool's 1-0 win over Stoke City.
Rodgers said Gerrard was a still a "world class player" but explained: "No matter how good a player has been once they come towards those final years then as manager you are having to think how you can get the best out of the player.
"Everyone knows his history at this club. What a player he has been, and still is. I see his talent every day and just need to manage that and think how it can be most effective. He is still a wonderful player with world class talent."
Rodgers also strongly denied any suggestion in Sunday newspaper reports there was a rift between him and his captain by saying "No, 150 per cent, no.
"I saw some things yesterday in terms of a rift between him and I which was unfortunate. The two words that would describe my relationship with him are honest and respect.
"To read yesterday about a rift is disappointing, but you can't control that" he said.
(Reporting by Mike Collett; editing by Justin Palmer)
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