Francis Ford Coppola is doing just fine after a hospital visit in Italy -- despite some alarming local reports ... TMZ has learned.

A source close to the 86-year-old director tells us he recently underwent a routine scheduled procedure with his physician of 30 years and is currently "resting nicely." The source says Coppola "appreciates everyone's concern."

While some outlets in Rome reported Coppola was hospitalized due to a mild cardiac arrhythmia before the procedure, we're told that's not true, and no emergency occurred.

Days before his procedure, Coppola was in high spirits at San Francisco's Palace of Fine Arts, where he hosted a discussion tied to his latest film, "Megalopolis." Coppola has been actively promoting the film -- his passion project starring Adam Driver -- with a cross-country tour in the States.

Despite some headlines, our source assures fans he's doing well and is continuing his work with the same passion that's defined his legendary career.