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.