Allardyce became England manager in July 2016 but was sacked by the FA after 67 days for behaving "inappropriately" when caught up in a newspaper sting.
The 63-year-old, who was filmed allegedly offering advice on how to "get around" FA transfer rules, has told the BBC in a documentary to air later this week that the decision to sack him was made "very hastily".
"Jealous is what I feel. Massively disappointed. That should be me there. Unfortunately because of the circumstances that wasn't the case," he added.
"...If a bit more time was taken, a bit more patience, I believe I should still be the England manager."
Southgate led England to the top of their World Cup qualifying group unbeaten and Allardyce gave his successor his full support.
"I've bounced back now - and while I will be extremely jealous I will be watching Gareth and the lads and hope they do extremely well," he said.
England begin their World Cup campaign against Group G opponents Tunisia on Monday, before taking on Panama and Belgium.
(Reporting by Christian Radnedge; Editing by Toby Davis)
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