The Three Lions will be managed by Thomas Tuchel, formerly of Bayern Munich, PSG, and Chelsea.
While Tuchel is rated as a top class coach, Allardyce believes the job should have gone to an Englishman.
"England is the best or one of the best jobs you can get. Even though I was only there a short time, it's still one of the best jobs you can get, you can't deny that if you're an Englishman," Allardyce said on talkSPORT
"The problem with the FA is they've gone against everything that they were doing in terms of building and developing their own from within.
"For me, that could have been Lee Carsley. If not, they could have gone all out for Eddie Howe or even Frank Lampard or Steven Gerrard in the end because you've got a quality squad."
"You have to be up there or thereabouts, I think, for anybody to consider you as the English national coach," he added.
"I think that's why I would have picked Eddie Howe and gone all out for it."