Allardyce is widely expected to be announced as Roy Hodgson's successor by the Football Association today.
Harper, 41, worked under Big Sam twice as a player with both Newcastle in 2007 and North East rivals Sunderland last season.
The former Mapgies keeper said he noticed a change in Allardyce at the Stadium of Light, with the 61-year-old being "calmer" and "more measured" than he was previously at St James' Park.
He told ITV News:
The second time he was calmer, more measured definitely. Obviously more experienced. But I think when he went to Newcastle the first time it was quite a big project and he tried to do probably a little bit too much, too quickly.
But he went into Sunderland pretty much by himself, had a look at the situation, assessed what needed doing and did it very well.
– Steve Harper
When asked if Allardyce would make a good England manager, Harper said:
I certainly think so yes. Organisation wise, structurally.
I think what was missing, certainly in the Iceland game, we saw an Iceland team who very knew their individual roles and responsibilities and the structure to play from. where as England didn't.
Now at Newcastle and then just before I went to Sunderland, and when I went there, he was playing three different systems. So he showed he's adaptive and he will need to do that because we do have an embarrassment of riches here in England with English players available.
Knowing how pragmatic he is, I'm sure he'll take his time to have a look at all of the players available and take his time to find a structure to give them a platform to produce the results we all want to see.
– Steve Harper
Harper spent 20 years at Newcastle before being released by the club in 2013. He then joined Hull City on a one-year deal that summer and was given a new one-year contract the following season.
After leaving Hull in 2015, Harper spent the first-half of last season as a free agent before joining Sunderland on a short-term deal until the end of the campaign.
Harper is currently a free agent and is looking to continue his playing career at a new club.