Good Hotels not far from trevi fountain.

Sandie, "convenient" is a relative term, and Trevi is no more than a walk-by: IMHO it's a minor attraction compared to many others. You're going to get opinions all over the map but we enjoyed our hotel in the Quirinale last time (we've been more than once) as it was very convenient to Termini and any number of attractions we wanted to take in. Trevi wasn't really very far away, although we only passed through there en route to others things. You also don't need to take cab there as the piazza is pedestrianized: you'll need to walk in. We were some distance from the Vatican and the Colosseum but that didn't bother us: we're stout walkers and public transit could fill in if needed.

Others on the forum prefer the Pantheon/ Piazza Navona area, or Trastevere for its atmosphere, although I wouldn't necessarily call it "convenient" depending on what you want to see.

Hotels often recommended by RS forum posters are:
Hotel Aberdeen (near Termini)
Hotel Sonya (near Termini)
Hotel Lancelot: (near the Colosseum)
Hotel San Carlo (near the Spanish Steps)
Hotel Oceania (near Termini)

If I was to venture a guess, you may be happiest in the Pantheon/Navona area, and Trevi won't be a difficult walk from there either. Just don't fixate on Trevi too much as you're not going to spend THAT much time there? Rome is best seen on foot whenever possible so unless you've mobility issues, try to do as much of that as possible? Really, historic Rome isn't big enough that much of anything is very far away from central hotels.

Do you have a per-night budget you'd like to stay within?

I haven't stayed near the Trevi Fountain, but you can check booking.com or another booking site for photos and reviews of hotels in the area by using filters or clicking on the map. It's within easy walking distance of most areas in the historic center, if you plan to see the fountain in the early morning or late evening.

We really enjoyed staying at Albergo Cesari during our last trip to Rome. It's just around the corner from the Pantheon (excellent gelato in the piazza, too) and a 3-minute walk to the Trevi Fountain. One of the highlights of the hotel besides the location and friendly staff was the hotel's roof-top - a wonderful place in the evenings to relax with a drink and enjoy lights. It's also where they serve breakfast.

Need to define convenient. And convenient to what? A cab ride would suggest being some distance away. Like someone else posted, Trevi Fountain, while magnificent, is just a ten minute visit and normally overwhelmed by tourists.

Don't know what your budget is, but here is a place you should check out. We have stayed at Hotel Fontanella Borghese many times and have always been pleased. While it is not "luxury", it is very nice and the location is great. The staff is very friendly and helpful. You can walk to Spanish Steps, Pantheon, Nuvona, even Vatican city. They also have a sister hotel closer to the Pantheon called Hotel Due Torri. We always send an email directly with our request for a booking and we get an answer back in 24 hours or less. Ask for a room overlooking the courtyard as rooms overlooking the side street can get noisy. The web site is www.fontanellaborghese.com.

A second vote for Albergo Cesari. We've stayed there twice and loved it both times. Five to ten minute walk in one direction to get to Trevi Fountain, five to ten minute walk in the opposite direction to get to the Pantheon. The desk clerks are very helpful. We loved the roof-top terrace. It's very reasonably priced for the wonderful location, if you can book it in far in advance, but you are already pretty close to your trip.

A lot depends on what you mean by "good." Good is not a very high benchmark in Rome. Good typically equals about average. Almost all the hotels closest to the Trevi are average, IMO. Without knowing someone's budget and taste, it's difficult to recommend a specific home-away-from-home.

There are about a hundred hotels within walking distance to the Trevi, and I mean an easy walk. You have the fabulous Pantheon on one side and the Via Condotti high-fashion on the other. You certainly won't require a taxi from either neighborhood. There are only a handful of hotels that are truly special in Rome. September is super high season, so the loveliest places will also be among the most expensive.

For some people, good means "good enough." For people like me, good means the "best." If you want a lovely hotel closest to Trevi, Grand Hotel de la Minerve is one I would consider.