According to Mike Klis of 9News in Denver, Cousins said, "I would hope any team is interested and would love it if Denver is interested."
Cousins played under the franchise tag for Washington in each of the past two seasons, but Smith's impending arrival is a sign that his time with the Redskins is over.
Based on his production as a starter over the past three campaigns, Cousins is likely in line to land one of the richest quarterback contracts in NFL history.
On Friday, Cousins also made it clear that he won't be close-minded as he prepares to hit the open market, saying, "I'm going to keep my eyes and ears open to anything and open. I'm in the information-gathering stage at this point."
The Broncos finished 5-11 last season due largely to the struggles of Trevor Siemian, Brock Osweiler and Paxton Lynch under center.
Denver is just two seasons removed from winning the Super Bowl, and its defense ranked third in total yardage allowed this season.
In addition to a championship-caliber defense, the Broncos have plenty of weapons on offense, including running back C.J. Anderson and wide receivers Demaryius Thomas and Emmanuel Sanders.
On Wednesday's episode of the Dan Patrick Show (h/t ESPN.com), Broncos pass-rusher Von Miller made it clear that the Broncos would benefit significantly from signing Cousins: "We need Kirk. I would like to have Kirk. We have great quarterbacks now. Kirk could take us over the edge."
The 29-year-old Cousins is a one-time Pro Bowler who is coming off a 2017 season that saw him throw for 4,093 yards, 27 touchdowns and 13 interceptions.
In each of the past three campaigns, Cousins has thrown for at least 4,000 yards and 25 touchdowns, and he led Washington to an NFC East title in 2015.
While Denver offers an enticing situation, the Broncos are far from the only team in need of a quarterback.
The New York Jets, Cleveland Browns, Buffalo Bills, Arizona Cardinals and Jacksonville Jaguars are among the teams that could be in the market for Cousins.
OddsShark lists Denver as the -150 favorite (bet $150 to win $100) to sign Cousins.
If the Broncos don't land Cousins, then they will be strong candidates to select a quarterback with the No. 5 overall pick in the 2018 NFL draft.