That intervention is significant as it provides the biggest hint yet that the 30-year-old is looking to continue his career with his boyhood club despite speculation about his future.
Gerrard and Reds vice-captain Jamie Carragher made the most of the time they had with Cole during England's disappointing World Cup campaign in South Africa to convince their international team-mate that Anfield was the right place for him.
Cole was a free agent having been released by Chelsea and Arsenal and Tottenham, who could both offer Champions League football this season, were also reported to be interested in signing the 28-year-old.
However, Hodgson said Gerrard and Carragher had done a good job in talking up Liverpool.
"Jamie Carragher and Steven Gerrard did a good job in selling the club to Joe at the recent World Cup," said the Liverpool manager.
"He was undecided about what he wanted to do and to be fair it takes a bit more selling to persuade a Londoner who has played all his life in London to move up to Liverpool.
"The simple solution would have been to stay in the London area."
Cole agreed personal terms on a four-year contract yesterday before flying to Liverpool's training camp in Switzerland to undergo a medical and meet his new boss.
"I am pleased to get the player because it is very important to try to improve our squad as much as we can in order to reach the high goals we have set ourselves," Hodgson told talkSPORT.
"It is also pleasing the reaction of the fans has been as positive as our reaction, so at the moment that is perfect.
"Now it is up to Joe to show on the field he is the right man."
Carragher felt the signing of Cole proved a point to their Barclays Premier League rivals who maybe thought the club were on the slide after a disappointing seventh-placed finish.
"It's great news and it is a message that Liverpool mean business," said the 32-year-old, who returns to Melwood today with fellow England internationals Gerrard and Glen Johnson to begin pre-season training after a three-week rest.
"We have signed a top international player who has been really important for Chelsea.
"A lot of clubs wanted to sign him but it is a real coup that he has chosen to come here and you have got to give the manager and the board a pat on the back for convincing him to come here.
"Every supporter that I have spoken to is absolutely delighted and all the players are the same - it's given everyone a boost, just at the right time.
"It has been a long summer and we have been in a transitional period but now the World Cup is out of the way, we can start to look forward again.
"The fact we have got him has given everyone a boost around the city.
"It's good news again and, hopefully, there will be more to come.
"It's the kind of signing that we needed to make."
Midfielder Lucas Leiva met Cole for the first time last night on his arrival at the team's base in Switzerland.
"He has a lot of experience and I have great confidence that he will help us a lot," he told the Liverpool Echo.
"He is a fantastic player and it is good that another English player is coming.
"As players we were looking to see signings like this.
"We have got Joe and Milan Jovanovic (Liverpool's other summer signing) has been really impressive in training as well.
"We all hope this means it will be a better season than last season."
Cole trained with his new team-mates in Bad Ragaz this morning and is due to undergo his medical this afternoon.