After signing quarterback Jake Plummer to a seven-year, $40 million contract, the Broncos attempted to trade Griese, but had no takers due to his huge contract.
By releasing Griese after June 1, the Broncos will save $6.3 million in salary cap space over the next two years - $4.9 million in 2003 and $1.4 million in 2004.
The Dolphins are expected to sign the son of Hall of Famer Bob Griese, the team's starting quarterback in three straight Super Bowls in the early 1970s.
At first it appeared the Chicago Bears, Baltimore Ravens and Dallas Cowboys would bid for Brian Griese, who was bestowed with a six-year, $33 million contract after the 2000 season. But the Bears gave a two-year, $5 million deal to Kordell Stewart, who lost his starting job in Pittsburgh, and the Ravens traded a second-round pick in April and a first-round pick in 2004 for the right to select California quarterback Kyle Boller.
The Cowboys showed some interest in Jake Delhomme, who signed with Carolina, but failed to even draft a quarterback. It appears new Coach Bill Parcells is ready to head into training camp with Chad Hutchinson and Quincy Carter.
That paves the way for Brian Griese to settle in at Miami behind starter Jay Fiedler, who has been beset by injuries the past two seasons. The Dolphins released backup quarterback Ray Lucas in April.
The successor to the legendary John Elway, who retired following the 1998 season, Griese never won over his teammates and coaching staff in Denver.
Griese, 28, has started 51 games the past four seasons. His most prolific year came in 2000, when he earned Pro Bowl honors after throwing for 2,688 yards and 23 touchdowns with four interceptions.
Denver made the playoffs in 2001, but Griese played just the first 10 games before suffering a season-ending shoulder injury.
Last season, Griese started 13 games, passing for 3,214 yards, 15 touchdowns and 15 interceptions. He missed two games with a knee injury and Coach Mike Shanahan chose to start veteran Steve
Beuerlein in another.
Receiver J.J. Stokes of the San Francisco 49ers and quarterback Akili Smith of the Cincinnati Bengals are also expected to be among the June cuts.
Stokes has spent his entire eight-year career with the Niners, but had just 32 catches for 332 yards and one touchdown in 12 games last season. Tai Streets surpassed Stokes as the team's No. 2 receiver and the Niners also drafted two receivers - Brandon Lloyd of Illinois in the fourth round and Arnaz Battle of Notre Dame in the sixth round.
Smith, who never lived up to his billing as the third overall pick in the 1999 draft, appears to be the odd man out in Cincinnati. The Bengals on Saturday signed veteran quarterback Shane Matthews in a move that likely signaled the end for Smith.
Celebrity Families of 2014 [PHOTOS]