Davenport suffered the injury in the Saints' 48-46 loss to the San Francisco 49ers on Sunday.
The 23-year-old was selected 14th overall in the 2018 draft and has stood out in his second pro year after a subpar rookie campaign.
Through 13 games, Davenport has the second-most sacks on the team (six).
The 10-3 Saints have already clinched the NFC South, though they lost the NFC No. 2 seed with the most recent defeat. New Orleans will try to retake home-field advantage on Monday against the Indianapolis Colts.