Tottenham Hotspur boss Harry Redknapp had been widely tipped to fill the vacancy at the end of the season.
However, it now appears that West Bromwich Albion coach Hodgson is the favourite to take the role that was vacated by Fabio Capello in February.
The anticipated outcome has been branded "cheap" and "safe" by former Southampton attacker Le Tissier.
"I think all the players, all the fans and all the media wanted Harry Redknapp to have the job," said Le Tissier. "It seems to me, as an outsider looking on, the FA has gone for someone who's going to do as he's told [and] isn't going to cost anything in terms of compensation.
"It looks to me like it's the cheap option and the safe option. Whether or not it's the right option only time will tell."
Hodgson is expected to meet with the FA later today.