The wiki confirms this is a real rumor. A June 8 article titled "Why Broncos Could Target Saints QB Spencer Rattler as Backup" is referenced across both Sean Payton's page and Saints QB Tyler Shough's page. So yes, there's been media speculation that Denver could pursue Rattler.
The logic makes sense: Payton spent 15+ years as Saints head coach, so he's deeply familiar with Rattler. Denver has only Bo Nix listed at QB on the fantasy-relevant depth chart (no named backup), and the Saints have moved on to Tyler Shough as their starter, meaning Rattler might be available. The article was published just a few days ago (June 8), so it's current buzz.
On the Ron Stewart show, clips note the Rattler/Shough QB situation in New Orleans — with Shough locked in as the presumed starter, Rattler's path to playing time is blocked, making a trade plausible for a contender like Denver needing insurance behind Nix.