There had been doubts over whether Gerrard, at 32, would be offered fresh terms when his existing contract expires in 16 months' time.
Rodgers though is quick to dispel any possible talk of a Merseyside exit for his skipper.
"He is vital to us, and I recognise that. 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," said Rodgers.
"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.
"Absolutely I would want him to stay beyond his current contract and extend it - there is absolutely no question about that.
"What I would say with Steven is he has a real hunger to succeed. He may be 32 years of age, but he's still got so much left."