A qualifying offer is worth $17.2 million and had to be presented to a team's free agents by Monday at 5 p.m.
Indians director of baseball operations Chris Antonetti indicated to Cleveland.com that the team will still try to sign both players, although other teams can begin making offers to the two players Tuesday.
Because they did not make those qualifying offers, the Indians cannot be compensated with drafts picks if Napoli or Davis sign with another team.
Napoli and Davis signed one-year contracts with the Indians last offseason, and they helped the Indians win the American League pennant for the first time since 1997.
Napoli, 35, set career highs with 34 homers and 101 RBIs during the regular season.
Davis, 36, platooned in center field with rookie Tyler Naquin. He hit .249 with a career-high 12 homers and led the American League with 43 stolen bases. He also hit a home run off Cubs reliever Aroldis Chapman to tie Game 7 of the World Series in the eighth inning.