Yes, big cats purr- some just do it a little differently. The four big cats who can roar (Lion, Tiger, Jaguar, and Leopard) have a slightly different hyoid bone construction than other big cats. The hyoid bone connects the tongue to the roof of the mouth and in these four cats, it is flexible- enabling their enormous roar to reach up to 120 decibels. But this feature also prevents them from purring on the inhale. They still can emit a sort of deep purr on the exhale though. In other cats, the hyoid bone is not flexible, so they can’t roar but they can purr- on both inhale and exhale. These include the mountain lion, the cheetah, bobcat, Eurasian lynx, and others.
There is one exception. The Snow Leopard does have the flexible hyoid bone but still can’t roar. It can purr though, on both the inhale and exhale.