Rome was a popular setting (not to mention travel destination) for American authors during the mid-19th century and into the 20th:

'The Marble Faun' by Nathaniel Hawthorne is the first book that comes to my mind when American fiction and Rome are considered together. Rome (the city, its buildings and monuments, byways and secret places) becomes almost another character in the novel and it is impossible to imagine this story set elsewhere.

Henry James used Rome as a setting for quite a few of his novels and stories, among them 'Roderick Hudson', 'Daisy Miller', and 'The Portrait of a Lady'. James visited Rome fairly frequently throughout his life and was very familiar with both the city and its flourishing Anglo-American community.

Roman Fever by Edith Wharton. Wharton loved Rome and a number of her other works also feature Rome as a setting for at least a chapter or two.

I'm sure other, lesser writers of the period probably used the Eternal City as a backdrop to their fiction as well.

I can't really think of much contemporary American fiction set in Rome. A bit of Carole Maso's 'AVA' takes place there, if I remember right...I'll have to ponder the question some more.