Gerrard has only 16 months of his deal left, and Rodgers is keen to remove any doubts that the England captain can play on for much longer at Anfield.
The 32-year-old has a guarantee of an ambassadorial role at his club when he ends his playing career, but there is an expectation he will play on for several more years following an exceptional performance in yesterday's victory over Fulham at Anfield, where he was one of the scorers along with Martin Skrtel, Stewart Downing and Luis Suarez.
With Gerrard's deal ending after the World Cup in 2014, it is essential that Liverpool tie up their captain for a longer period.
"Steven is vital to us, and I recognise that," Rodgers said. "I know he has 16 months left on his contract, but I would want him to go beyond that for this club and this team.
"This is a guy who has led his life right as a football player, and that allows you to play well into your 30s. Absolutely I would want him to stay beyond his current contract and extend it – no question about that.
"He has been an inspirational captain for us, he has been a huge figure in the development of this team. We are improving, we are growing, and he has influenced that."
The Liverpool hierarchy is cautious about handing out deals to players in their 30s, with Gerrard's most recent contract, signed a year ago, amounting to just a one-year extension.
Jamie Carragher's contract ends at the end of the season, although the lack of negotiations on his future has been mutually agreed until the player makes a firm decision on his intentions and the club informs him what role, if any, they see for him.
Gerrard, who has played every minute of Liverpool's Premier League season, showed against Fulham that he still provides a dynamic presence.
"You look at his stats this season and he is right up there as one of the most influential midfielders in the Premier League," Rodgers said.
Sturridge passed a medical on Merseyside yesterday after negotiations were resolved over the weekend. Liverpool hope to announce his arrival soon after the transfer window opens on January 1.
He could be one of several comings and goings, but Downing is hoping that he has done enough to save his Anfield career after learning from the "kick up the backside" he received from Rodgers.
Downing's ability has never been doubted – the list of managers who have raved about how good he is in training is long – but it is the ability to consistently take this on to the pitch that has raised questions about a fragile character.
The Downing gags at Anfield are as predictable as those in your festive crackers, mild chuckles as he claimed his first assist of the season with a defence-splitting pass to Gerrard and similar mirth when he slammed in his first league goal for the club early in the second half.
In his defence, these statistics would not be so damning if Liverpool had taken numerous chances Downing created last season and he had not developed an infatuation with the woodwork.
Nevertheless, a £20m outlay 18 months ago ranks among the most excessive of a profligate era, so after launching a rocket of his own with his left-footed screamer, it was heartening to hear Downing in similarly robust form off the pitch minutes after the final whistle.
"The manager was pleased with me at the start but then things changed very quickly," Downing said.
"I don't know who makes that decision but one minute I was told I was doing well and the next I was told I could go, so it was a bit confusing.
"If it was meant to be a kick up the backside, then maybe it has worked. I have worked hard to get to a club like this and I don't want to leave in a hurry."
Rodgers acknowledged the possibility of an about-turn, although he will also tell the player a standard has been set.
Downing needs to produce this on a weekly rather than occasional basis. The same applies to his team.
After the most worrying spectacle of Rodgers' Liverpool reign a week ago, this was the most reassuring. Gerrard was at his dynamic best and Suarez in excellent form until he scored in the closing stages.
Fulham were abysmal, but worse teams have taken points here in the past 18 months. Liverpool are now eighth. They will be treating fans for vertigo soon. (© Daily Telegraph, London)