A seventh-round stoppage of Deontay Wilder in Las Vegas on Saturday ensured Fury won the WBC belt and became a world heavyweight champion for the second time in his career.
The 31-year-old previously beat Wladimir Klitschko in November 2015 but vacated the IBF, WBA and WBO titles while he fought depression and drug addiction.
He made his return to the ring in June 2018, shedding more than seven stone in six months, and completed his remarkable turnaround by beating Wilder two and a half years later.
"I just want my son to retire now," John Fury told Good Morning Britain.
"He's done enough. It's been an uphill battle for him.
"I think it's in the back of his mind. He can't do any more. He's won every professional title. Enough is enough. There's more to life now. He's given it his all.
"He's got no more to prove."
— World Boxing Council (@WBCBoxing) February 23, 2020
Fury has previously said he would "seriously think about walking away" once his current three-fight deal expires.
There are two fights remaining on that contract and one of those would be a third bout against Wilder if the American exercises a rematch clause to fight Fury again.
Saturday's second fight captured the attention of US President Trump, who suggested to reporters he would invite both men to the White House.
"That was a great fight," Trump said.
"Two great fighters, really very exciting. Maybe we have to bring them both to the White House because that was really a good one.
"In fact, I think we'll do that."
Some sports stars have swerved invitations to the White House since Trump assumed office yet Fury Senior encouraged his son to go.
"That's good for a Fury, isn't it," he said of Trump's offer.
"I'm a big fan of Donald Trump. It's been an amazing journey, look where it's ended.
"And what a great point to bow out on - a meeting in the White House."
Photo Getty Images Caption: Tyson Fury