Allardyce claims he was approached by the Magpies after Rafael Benitez opted not to renew his contract at the club.
Steve Bruce later left Sheffield Wednesday for the role, with Allardyce unwilling to take on a second spell at Newcastle.
But the 64-year-old, out of work since departing Everton in May 2018, indicated he is willing to consider a comeback.
"It didn't happen last season which was my first year out of professional football for the first time since I was 15," Allardyce told Sky Sports.
"It was a bit strange just watching all the games but I found it to be very interesting in seeing how I feel about the game and how I would feel about what I would do if I got back in – you have plenty of time to get ready if you do want to get back into the game."
Bruce led Newcastle to an eye-catching win at Tottenham prior to the international break and Allardyce feels he was the right man for the job.
"Newcastle wasn't for me," Allardyce told Sky Sports. "I was very chuffed that Mike [Ashley, Newcastle chairman] considered me again but I felt that Brucey was a better fit than me and good luck to him.
"It is his dream move to manage his hometown club and I wish him well. It was the only thing that came up this summer."
Allardyce's former clubs include Crystal Palace, Sunderland, West Ham and Bolton Wanderers, while his brief spell in charge at St James' Park lasted from May 2007 until January the following year.