Cook and stir sausage in a large pot until browned, about 5 minutes. Stir in onions, mushrooms, and garlic. Pour in chicken broth, tomato sauce, and diced tomatoes; bring soup to a boil.
Stir pasta into the soup. Cook, stirring occasionally, until almost tender, about 10 minutes. Stir in spinach, mozzarella cheese, and Parmesan cheese. Remove from heat; stir in basil and let stand until pasta is tender, about 2 minutes more.