Talks have been ongoing and have continued to gather traction, but there is no deal imminent, sources said. There is currently a moratorium on trades as the NBA goes through procedures to start next season ahead of the league's draft next week.
The Suns have a promising, young roster led by Devin Booker and Deandre Ayton. The team finished last season strongly by going 8-0 in the NBA bubble in Orlando, Florida, narrowly missing the play-in tournament.
In addition to their young roster, contributing to the Suns' momentum is the near completion of the first phase of a $230 million renovation of Talking Stick Resort Arena. The Suns also are set to open a new, state-of-the-art, $45 million practice facility this month.
The Suns could have in excess of $20 million in salary-cap space going into the transaction season, leaving the possibility that they could assemble a trade and still look for other free agents.
Paul played a season under Suns coach Monty Williams in New Orleans from 2010 to 2011, and he has spoken highly of Williams in the past. Paul has also praised the Suns' style of play.
"His basketball mind ... I used to get so excited before the games about the plays he was drawing up," Paul told reporters before facing Williams' Suns in December.
Paul has two years and $85.5 million left on his contract, which is a motivation for the rebuilding Thunder to consider trading him. The Thunder, however, have told teams that they remain interested in keeping Paul for next season.
Paul is coming off a season in which he averaged 17.6 points and 6.7 assists as he led the Thunder to the playoffs. After the campaign, he was named second-team All-NBA.
ESPN reporter Adrian Wojnarowski contributed to this story.