What settings? What FPS? If you want to game with good FPS and high settings on multiple 1080p displays you need something like a 7970. With the rest of the system being high end top. The 6570 isn't even really meant for gaming.
The lowest Toms recommends is the 6670, and they don't recommend it for 1080p. It would get about the sane FPS you get now with ONE monitor.
www.tomshardware.com/reviews/gaming-graphics-card-review,3107-2.html
Yes technically a 6570 can do eyefinity, and in some very low end games like racing/flight Sims it would probably work. But multimonitor for high end triple AAA gaming is EXPENSIVE. A 6570 is probably going to have lower FPS than your current cards on a SINGLE monitor. Add a second monitor and it will get cut in half.
with a 6670 you MIGHT get 20 FPS average on low, doing down to 10. It will be unplayable and again you wont be able to even play the game because of the Bezel. What you want to do with your apparent system/budget does not make any sense and I would not at all suggest it.