Closest Catholic church celebrating Mass to the Roman Forum

I'm sure there will be service at Santa Maria in Aracoeli, near the Campidoglio. But there are so many churches within 5 min walk, that is just a matter of checking their schedules.

Santasusanna.org is the American Catholic Church parish in Rome, their website may have Mass schedules or you can email them.

There are more than 900 churches in Rome. If you don't care if the mass is in Italian, once you're there, just look at the little sign on the door of one of the many churches all around there.

If you there's an early mass near your hotel, then you could do both activities.

Yes, Santa Maria in Aracoeli on the Campidoglio would be close, and it looks like masses are at 8:00 and 12:00. It's a bit of a climb up to this one so do some reading in advance:

It would be helpful to know where you're staying, though, as will very likely be some good alternatives along the way?
As someone else mentioned, there is a church on nearly every corner in Rome, and masses on Sundays are so frequent at many that sightseeing the best of them is more easily done on other days of the week!

Two that are right adjacent to the forum are
--S Francesca Romana, which is at one end of the forum.
--Saints Cosmos and Damien (probably something like S Cosma e Damiano in Italian); this church is on the site of the temple of Romulus, and ruins of the temple are visible in an area at the end of the nave. My recollection is that it also has a lovely early apse mosaic. This is the one of the two that I would choose. You have to enter from the street, so takes a little bit of effort to find the entrance.