The fight broke out between a 19-year-old Dutch teen and a 44-year-old American woman who wanted to take selfies at the same spot in front of the fountain at the same time.
They had an exchange of words before a physical fight began, and their family members joined in.
According to the local newspaper La Repubblica, there were eight people involved in the fight before four police officers intervened.
The tourists have been charged with violence.
To protect the city's fountains, Rome has been imposing fines of up to 240 euros (about 274 US dollars) for bad behavior, including picnicking or camping around the fountains, bathing their feet in the water or swimming in the fountains.
Now the city council is considering a plan to shuttle tourists past the Trevi Fountain in a single file to prevent large groups from gathering around the tourist attraction.
The Trevi Fountain is the largest baroque fountain in Rome and one of the most famous fountains in the world. It has appeared in several notable films, including La Dolce Vita and Roman Holiday.