Gerrard, the influential Reds and England skipper, has claimed winners' medals in most of the major competitions open to Liverpool but the Barclays Premier League is a glaring exception.
With the Merseysiders finishing eighth last term and the team being rebuilt by Rodgers, Eriksson has suggested that, at 32, Gerrard is running out of time to fulfil his league dream.
But Rodgers said: "I think Steven, over the years, has had a number of chances to leave here.
"He has always felt this is the club where he could still win it and he has still got a number of years left in his career.
"We understand how difficult it is going to be but I think he will still feel the dream is here. I don't think he needs to move anywhere.
"Of course we will go through difficult moments - but his dream is to win the Barclays Premier League with Liverpool.
"I, along with everyone else and the supporters, will fight for our lives to try to make that possible.
"He is an iconic figure for this football club. His status and everything he has gained in the game has been through Liverpool Football Club.
"He has won every major honour you could possibly win at this club, but obviously there is one missing and he will want to fight for the rest of his career to win it here.
"Thankfully with Steven, he is realistic as well. He knows the reality. He is not misguided with anything.
"Sven obviously has his reasons for saying that but I have been delighted with Steven's contributions since I have been here.
"I have seen him from the outside but from the inside working with him as a man, as a player, he is a wonderful role model for everyone here.
"Let us see if we can fight to win it ourselves and hopefully he can stay here and be a real ambassador for the club."
Rodgers is considering making changes to his squad in the final two days before Friday's transfer deadline.
The club have been linked with a number of players in recent days, with Arsenal and England winger Theo Walcott the latest high-profile example.
Rodgers, speaking at a press conference to preview his side's Europa League qualifying round play-off second leg against Hearts, was unwilling to comment on speculation.
He said: "There is nothing on any player. I will speak about a player after Friday if he is in the door.
"In these next 48 hours speculation and gossip will be rife, I could be here all day answering speculation. I have no need.
"We have got the end of the window coming up. I want to work with a tight-knit group between now and the end of January and then we can assess it from there."
There is also uncertainty over the futures of a number of the current Liverpool squad, including Andy Carroll, Charlie Adam, Jordan Henderson, Joe Cole and Jay Spearing.
Rodgers said: "We have had a number of enquiries about some of our players.
"I have been open and up front with all the players. You can see from Sunday's game our spirit is very strong in the group, but the reality is players want to play.
"Everyone knows where they stand at this moment. One or two of them may have decisions to make as to whether they want to be a squad player or play regularly somewhere else."
After crashing 3-0 at West Brom in their Premier League opener, Rodgers hopes his side's pulsating 2-2 draw with champions Manchester City on Sunday will kickstart their season.
Their most immediate task is to claim a place in the Europa League group stages by completing the job against Hearts after claiming a 1-0 aggregate lead from the first leg at Tynecastle last week.
The likes of Gerrard and Luis Suarez were rested for the trip to Edinburgh but could feature despite the proximity of Sunday's home clash with Arsenal.
Rodgers said: "It is easier when you have two home games. With the travel in that last week and the recovery process, it is easier to go Thursday-Sunday with a reasonably strong side.
"Tomorrow [Thursday] we'll be strong. We're looking to win the game and keep the momentum going.
"Our performance levels at the weekend were excellent and we want to just keep that going forward."