Former captain Beckham has offered his services to the Three Lions coach and claimed that he would be "very proud" to be involved in his backroom staff.
"Someone of David's stature, with his qualities and personality, I'm 100 per cent certain if he can find the time to lend his presence to the FA they would be happy to find a role for him," Hodgson told Sky Sports News.
"If he could make himself available for England in some way or another I have no doubt in my mind that [FA chairman] Greg Dyke and [FA general secretary] Alex Horne would welcome his presence within an England capacity."
Beckham made 115 England appearances in an international career that spanned 13 years.
England get their Euro 2016 qualification campaign underway with a trip to Switzerland on Monday.