Under NBA rules, contracts cannot be finalized until Friday but news about player plans began circulating quickly once the period for conversations began.
George, considered a possible target for the Los Angeles Lakers in a bid to lure Cleveland star James, was captured on video at a party at teammate Russell Westbrook's home saying he would stay in Oklahoma City.
"I'm here to stay," George said in the posting by a reporter for "The Franchise" radio station.
George later posted a photo of himself with Westbook on Instagram with the caption "Unfinished Business" and the Thunder tweeted that general manager Sam Presti "intends" to sign George, a deal ESPN reported was worth $137 million over four years.
Carmelo Anthony had opted last week to remain with the Thunder, which lost to Utah in the first round of this year's NBA playoffs.
Also citing "unfinished business" in staying put was Houston Rockets star point guard Paul, who tweeted "UNFINISHED BUSINESS... RUN IT BACK," with an emoji of a Rocket ship blasting off.
The Rockets, who had the best record in the NBA last season at 65-17, lost to eventual champion Golden State in seven games in the Western Conference final, with Paul out with an injury the past two games.
The Houston Chronicle and ESPN reported Paul's deal to reunite with NBA Most Valuable Player and scoring champion James Harden will be for four years at $160 million.
Kevin Durant will re-sign with Golden State for two years, the New York Times reported, keeping the two-time NBA Finals Most Valuable Player with a club that has won three of the past four crowns and seeks a fifth consecutive trip to the finals next year.
Four-time NBA MVP James, who has played in the past eight consecutive NBA Finals, remains the biggest name up for grabs, although ESPN reported the Cavaliers were on the phone with James just seconds after midnight to express their desire to keep him with Cleveland.
The Cleveland Plain Dealer and ESPN reported representatives of James and the Philadelphia 76ers had a Sunday session scheduled in Los Angeles.
While the Cavaliers can offer more money and longer contract terms, the Lakers have greater salary cap room and could land another star player to join James in Los Angeles to better build a championship contender.
And the 76ers offer talented young teammates such as NBA Rookie of the Year Ben Simmons of Australia and Joel Embiid from Cameroon.
On Friday, Simmons posted an Instagram message from a game featuring himself and James with the message "KJ x FP" figured to mean "King James" times "Fresh Prince", a nickname Simmons has adopted recently.
Simmons and James are represented by the same agent.
Embiid tweeted, "The Lakers are FOREVER gonna be Kobe's and Magic's team.... Process that," citing the long-lasting legacy of retired stars Magic Johnson and Kobe Bryant with a nod to the Sixers' "Trust the Process" motto.
The Lakers reportedly had been hoping to swing a trade with San Antonio for unhappy star Kawhi Leonard and sign George to make a pitch to James.
The Lakers have contacted New Orleans center DeMarcus Cousins, ESPN reported Sunday.