As with all banks, where they pull is primarily based on where you live. So no one here can answer for you for sure (unless they live very close to you).

All I can say is that for me in SoCal, they pull TU. But I don't know if that's the case for all of California.

Have you applied for any Barclay card before? If so, chances are they'd use the same bureau for you as they did before.

But if not, you have to search online for info on which bureau which banks pulls by state (and it may even more finely detailed than on a whole state basis sometimes).