Alex Horne, one of the four men who appointed 66-year-old Hodgson as England boss, told Friday's edition of The Times that he could not see any reason to make a change before Euro 2016.
"For me personally, Roy is our manager to France and I'm happy with that, very happy with that," he said. "I think he's been brilliant. He's exactly what we hoped for."
Asked what would be an acceptable performance at the World Cup in Brazil, where England face Italy, Uruguay and Costa Rica in Group D, Horne said an unbeaten qualifying campaign had already proved Hodgson's value.
"Pass mark? For me, personally, he's done it by getting through that group," said Horne.
"We go with confidence. We'll prepare well. We go optimistic that we are going to enjoy it and play well. It's not an easy group, it's not an easy tournament."
The experienced Hodgson, who led Switzerland to the World Cup in 1994 and Euro 96, took over the England job in May 2012 after Italian Fabio Capello resigned and steered the country to the Brazil tournament from a group including Ukraine and Poland.
"Roy stabilised the team and he's qualified us well in a tricky group," said Horne. "He kept his head convincingly, won the last two (matches against Montenegro and Poland). I've huge respect for him."
Horne also told The Times the FA had made a bid to UEFA to host the first game and opening ceremony of Euro 2020, which will see matches spread around Europe for the first time.
He said he was confident Wembley would get three games - two in the group stage and a quarter-final - and was also pushing European soccer's governing body for the opening match.