I thought all museums are free on Sundays. Does that mean the Borghese is or should I make reservations and pay the euro13
I'm not sure why you thought that, but it's not true. And, at any rate, when they do have free days, they're a zoo.
I highly suggest making a reservation. When I went, a couple of years ago, entry is by the time on your ticket. You enter in as a group at that time and they shoo you out before letting the next group inside. I don't remember how much time you get but we were so dazzled by the marble statutes and their setting that we barely saw anything of the upper floor paintings. It is a very special museum and one of our favorites in Rome.
Only the state run museums are free on the first Sunday of the month. The Borghese is one of those but you still need to book a reservation. You need to call to book from free Sundays, you can't do it online. All other Sundays the museums are regular price.
Donna
The local Italians love a deal as much as any American tourists and there are a lot more of them than us. Free Sunday is an absolute zoo and I would avoid it. Borghese always requires a reservation.
And since the Borghese always limits the number of people, free Sunday is the best time to go.