"I said this a while ago: Colin had his chance to be here. We offered him a contract. He didn't take it," said Elway, per the team's official website. "As I said in my deposition -- and I don't know if I'm legally able to say this -- he's had his chance to be here. He passed it."
The legal reference related to Kaepernick's collusion grievance against the NFL. Kaepernick filed the grievance under the collective bargaining agreement against the NFL in October, accusing all 32 owners of colluding against signing him to an NFL contract.
Elway and the Broncos were unhappy with the team's quarterback play in the preseason opener against the Minnesota Vikings last weekend.
Case Keenum, who signed a two-year, $36 million deal with Denver in the offseason, struggled against the Vikings, completing just 1 of 4 passes for five yards.
"I was hoping to get off to a better start than we had the first week in preseason, but it's been a good week of practice and Case has had a good camp," said Elway. "Hopefully, we dig in a little bit more this week and get off to a better start."
Paxton Lynch, a first-round pick of Denver in the 2016 NFL Draft, also labored in the exhibition opener and was dropped to No. 3 on the depth chart after he completed just 6 of 11 passes for 24 yards with an interception.
"It's the same role, it's just reversed," Elway said of Lynch's demotion to third-stringer for Saturday's game against the Chicago Bears. "I hope Paxton takes advantage of it and goes out and plays well."
Chad Kelly, taken with the final pick in the 2017 NFL Draft, was bumped up to No. 2 on the depth chart after completing 14 of 21 passes for 177 yards with two touchdowns and one interception against the Vikings.
"I had a feeling he was going to play well. He's a competitive guy, he's played a lot of snaps and he's instinctive," said Elway. "He took advantage of the opportunity that he had. He was looking forward to going out and playing and chomping at the bit. It did not surprise me, the way Chad played. He played very well. He's got good instincts. He deserved the chance to play with the twos this week."
Elway told reporters Thursday that he has a simple criteria for what he's seeking in a backup quarterback.
"Somebody that can win football games. I think that's the key thing," Elway said.