"It's a lot of media hype," a Cowboys source told Calvin Watkins of The Athletic. "Seattle [is] not giving him away and we're not overpaying."
Thomas posted a message on Instagram this week asking the Hawks to either give him a contract extension or trade him elsewhere.
"Extend .....if you don't want me let's make a trade happen. I understand it's a [business]," he wrote.
In early July, Bryan Broaddus of the Cowboys' official website said on 105.3 The Fan radio (via the Dallas Morning News) that a possible Thomas acquisition "still is in play" for Dallas.
Thomas was previously caught on camera after a December game telling Cowboys head coach Jason Garrett to "come get me":
The 29-year-old University of Texas product has one year left on his contract, which carries a cap hit of $10.4 million in 2018, per Spotrac.
Thomas, a six-time Pro Bowl selection and three-time first-team All-Pro, racked up 88 combined tackles, seven passes defended, two interceptions and a forced fumble in 14 games last season. Pro Football Focus graded him as the league's ninth-best safety.
He'd represent an upgrade over the Cowboys' projected starter, Jeff Heath, whom PFF rated as No. 35 last year.
Dallas is running out of time before the start of the new season, however, and it doesn't sound like there's much concrete progress toward a trade despite the rumors.