"We've offered Elvis a contract," Broncos executive vice president of football operations John Elway said Tuesday. "We'd like to have Elvis back."
Elway did not delve into specifics of the offer.
Dumervil and the Broncos agreed upon a restructured contract that would have paid him $8 million in 2013. However, Dumervil's agent did not get the paperwork faxed to the team until seven minutes after the NFL deadline on Friday. The Broncos were forced to cut Dumervil because they would have had to pay him $12 million in 2013 otherwise.
Dumervil fired agent Marty Magid on Saturday. The Broncos are open to re-signing Dumervil and he reportedly wants to return. But there are salary-cap issues at play. Baltimore and Pittsburgh also have reportedly expressed interest in the 29-year-old pass-rusher.
Dumervil signed a six-year, $61.5 million contract in 2010 when Josh McDaniels was coaching the team and before Elway joined the Denver front office.
Dumervil had 11 sacks last season, second on the team behind Von Miller. Dumervil led the league with 17 sacks in 2009.
ESPN.com's Bill Williamson contributed to this report.