I've stayed at two convents/monasteries in the areas you mentioned. The following convent is one-half block from SM Maggiore and about four blocks from Termini. It's unlikely that you'll find anyone who speaks English, but with a little creativity I've never had trouble communicating.
Suore di Santa Elisabetta
Via dell'Olmata, 9, 00184 Roma, Italy
+39 06 488 8271
The Camaldolese Benedictine monastery of San Gregorio Magno is about two blocks from the coliseum and across a very busy street from the Circus Maximus. It's a community of men, and consequently is a little less warm and fuzzy than a convent. When I was there, no one spoke English. But they elected an English-speaking prior in the last couple of years, so language may be a little easier now.
Both of the above are wonderful historical buildings. You can check out reviews on trip advisor.
Also the website monasterystays.com lists many convents and monasteries, and they'll book them for you. The cost is a little higher than if you do it yourself.
Finally, the website of the American Catholic parish in Rome, Santa Susanna, lists convents and monasteries that accept guests.