Floyd sustained the injury during the Bears' 24-23 victory in Saturday's preseason game against the Denver Broncos.
The Bears hope Floyd will be ready to return for the Sept. 9 regular-season opener against the Green Bay Packers.
"We're staying optimistic for Green Bay," Nagy said, according to ESPN.com.
The injury may not be fully healed by then, but Floyd might be able to play with protective covering on the hand.
Nagy said the injury included a fracture on Floyd's index and middle fingers. The injury occurred on a cut block late in the first quarter. Floyd injured his hand as he braced himself for the fall.
"Just unlucky," Nagy said.
Floyd, the ninth overall pick in the 2016 NFL Draft, has been plagued by injuries, limiting him to 22 games in his first two seasons.
Considered the team's top edge rusher, Floyd has 11.5 career sacks, including 4.5 sacks last season when he played 10 games, all of them as a starter.
Despite missing six games, Floyd was second on the team in sacks last season, behind only Akiem Hicks, who had 8.5 sacks.
Floyd's availability is crucial for the Bears, who don't have much depth at his pass-rushing position.