No, American English is pretty distinct from British English at this point, especially what most people think of as a Southern dialect. The closest dialect of American English to British English (I assume you mean BBC British, because British dialects get more disparate the farther down the social ladder they go) is probably something in New England very near the East Coast, or maybeupper-class coastal Southern English. The problem is that the accents have remained more similar than the dialects have.