Chicago also claimed right-hander Cory Mazzoni off waivers from the San Diego Padres on Monday. Outfielder Leonys Martin was assigned to Triple-A Iowa, and the Cubs selected pitcher Matt Carasiti's contract from their top farm club.
Arrieta, who won the NL Cy Young Award in 2015, went 14-10 with a 3.53 ERA in 30 starts this year.
Davis, who was acquired in a trade with Kansas City in December, had 32 saves with the Cubs and was an All-Star for the third straight season.
Arrieta and Davis have until Nov. 16 to decide on their $17.4 million, one-year qualifying offers. Both players are expected to decline the proposals.