The 62-year-old met the England midfielder and vice-captain Jamie Carragher on his first day at the club and despite concern over Gerrard's future Hodgson said the player did not even raise the issue.
"Obviously Steven and Jamie are the playing heartbeat of the club and it's very important we keep people like that with us," he said.
"I anticipate there might be a situation where other big clubs will try to sign Steven but he gave no indication that he wanted to leave."
Hodgson told LFC Weekly: "Nobody at the club wants him to go so I will be doing my utmost to make sure he stays. I am confident he will."
Hodgson has not yet had a chance to meet striker Fernando Torres, who is also being linked with a number of top European clubs, because he is still on duty with World Cup finalists Spain.
However, the Englishman remains optimistic he can convince the 26-year-old that things will improve at Anfield and he can achieve his ambitions with the Merseysiders.
"Fernando is a wonderful striker and I understand the fans' concerns about his future," said the former Fulham boss.
"Unfortunately with the World Cup still on I won't be able to meet him face to face for another few weeks - that's just circumstance and you won't hear me complaining about it.
"When I do meet him I will be doing everything I possibly can to convince him that Liverpool is the place to be.
"If he sees progress I am confident that he won't want to go anywhere else. I think he understands the club and the city."
Hodgson is also keen to engage club legend Kenny Dalglish in a more active role.
The former player and manager has been working at the club's academy in an ambassadorial role and was part of the selection process to find a successor to Rafael Benitez.
Dalglish put his name forward to return to a position he successfully held in the late 1980s and early 1990s but that was rejected by the board and doubts were raised about whether he would sever ties with the Reds as a result.
However, Hodgson is a long-time friend of the Scot and wants to embrace the knowledge and influence the 59-year-old possesses with a more formal position in youth development.
"I never doubted that Kenny would leave. He's been offered a role, which I believe is a crucial role for the club - not just in the short term but in the long term," Hodgson added.
"He is very aware of all the players at the academy and I understand there are a few promising players emerging.
"He will liaise between Melwood (the first-team training ground) and Kirkby (the academy) and keep me updated with any developments that he deems necessary.
"There is a bridge between the academy and the first team that needs to be created and I believe he can help build that bridge."