The 29-year-old defensive back is scheduled to make $7.8 million in 2019 and can become a free agent next offseason, per Spotrac.
Klis reported Sunday that Denver general manager John Elway and Harris' agent, Frederick Lyles Jr., talked over the holiday weekend.
Harris requested a new contract or a trade in April as he entered the final year of a five-year, $42.5 million extension he signed in December 2014. Elway had said he didn't "have time right now" to address Harris' situation as his focus was on the 2019 NFL draft.
Harris, though, never ruled out the possibility of reaching a new deal with the Broncos and made it known that he hopes to play out his career in Denver.
"Oh yeah, I always wanted to retire here," Harris told The Athletic's Nicki Jhabvala on April 24. "I've been patient. I'm still open to coming back. I'm never closing that door until they close it. ... We could still get a deal done with the Broncos. Oh yeah. It's not over yet."
Harris declined to say whether he would hold out if neither of his requests was met, only telling Jhabvala that "it shouldn't get to that point."
An undrafted free agent in 2011, Harris has spent his entire eight-year career in Denver. He has earned four Pro Bowl nods and was a first-team All-Pro in 2016. He was part of the Broncos' vaunted "No Fly Zone" secondary that led the team to Super Bowl 50 victory during the 2015 season.
According to Pro Football Focus, nobody in football has been better in the slot than Harris in recent years:
Harris is coming off a Pro Bowl year in 2018 that saw him pile up 49 tackles, three interceptions, 10 pass breakups and one sack.