"Of course I would like to be England captain," Gerrard said at Liverpool's kit launch.
"It is the kind of thing every player dreams about. But I am not the kind of player to talk about it and try and get it that way.
"The decision is down to (England manager) Steve McClaren and if it comes my way then great -- if not, whoever gets the job will have my full support."
Gerrard is a leading contender for England skipper along with Chelsea's John Terry and Frank Lampard.
Beckham stood down as captain earlier this month after England were knocked out of the World Cup in Germany in the quarter-finals.
The Liverpool captain said he was fully rested after the World Cup and was looking forward to helping Liverpool challenge champions Chelsea for the title.
The signings of Craig Bellamy, Gabriel Paletta, Mark Gonzalez and Fabio Aurelio in the close season has increased confidence at the club over their ability to take on Jose Mourinho's side.
Liverpool won the FA Cup last season and the Champions League in 2005.
"I like to think this club is on the brink of something very special at the moment," the 26-year-old added.
"There are a lot of players in our dressing room who are only missing the Premiership title from their CV.
"We all have a big determination to push on and I am keen to hold that trophy over my head," Gerrard said.
