Neither is correct. We say "in [the] coming days", but never "on coming days." For example, "In the coming days we will have more time to exercise." Another example, "There will be an opportunity to study English in the coming days."

Undoubtedly in coming days

"in" We use "in" for spans of time longer than 1 day. Since "coming dayS" is longer than a day, we use "in". Very rarely, you might see "coming" used synonymously with "upcoming", in order to refer to individual days in the future, and in those cases you would use "on": "I will be absent on exactly 5 (up)coming days in the coming 3 months."