Byram is a product of the Leeds academy and is currently in his second stint at Elland Road after rejoining the club last summer on a free transfer.
The defender initially left Leeds for West Ham in January 2016 in acrimonious circumstances after refusing to sign a new contract.
Then Leeds owner Cellino accused the player of being too big for his boots for refusing to sign the new deal, but Byram provided another side to the story.
He claimed that the Italian was trying to convince him to sign a new contract on reduced terms while West Ham and Everton were offering Premier League football.
Byram admitted that he regrets not playing a part in Leeds getting promoted to the Premier League a few years later, but insisted that he is prepared to right that wrong this time around.
The defender told The Athletic: “I’m not sure what I am allowed to say, but Massimo Cellino tried to convince me to sign a new contract on much worse terms.
“When West Ham and Everton at the time were interested, giving me the chance to play in the Premier League — whether you’re a Leeds fan or not, given the opportunity to live your dream, the top aspiration, most people if they’re honest would have taken the chance.
“But I’ve thought back since and with hindsight, if I’d known that the club would achieve promotion [in 2020] — to be able to play in the Premier League with Leeds — I’d like to say I’d have turned the move down.
“You just can’t guess what the future holds.
“You have to make a decision based on the facts, on what you know and what you’re given.
“There are some regrets that I missed out on getting Leeds promoted, but I feel like this is a second chance to achieve that.”
Byram was released by Norwich City at the end of last season and earned a contract at Leeds after training with Daniel Farke’s squad during pre-season.
