The two airports have been shut to passenger flights since March 13.
Italy’s current coronavirus lockdown expires on May 4, when many restrictions remain but more people are expected to head back to work and travel within regions will be allowed under some circumstances.
Over a series of weeks, more businesses will be allowed to reopen.
The nationwide quarantine began on March 9.
The opening of the airports next week will allow for testing of a screening system for coronavirus, the ministry said, without elaborating.
Italy’s trains will also add new long-distance connections “in order to ensure minimum essential services,” it said.