Hull boss Brown worked as Allardyce's assistant at Bolton Wanderers and told the Evening Chronicle: "Sam was offered the Newcastle job two years ago but turned it down.
"Being from the North East and recognising the size of the club, I couldn't believe why he would say no.
"Now he's got the club of any manager's dreams."
Allardyce insisted he was no longer interested in the England job when he arrived at Newcastle.
But Brown believes that, if Big Sam is a success on Tyneside, he will again be linked with the national team role.
He went on: "The one thing they say about the England job is that you need the ability to handle the Press, and to do that you have to be strong and forthright to the nth degree.
"It's just about being honest to yourself, and that is something Sam always is.
"I firmly believe that Sam should have got the job first time around, but now he's got to succeed at Newcastle and prove himself.
"If he does that, he will get another stab at the England job - it's as simple as that."