There can be a small effect of the supply on the noise. Which way and how much depends on the OP type. It can be either way and normally the effect is rather small, especially with newer types. Some older type may have a not so stable current biasing and this way can have a decreasing noise with increasing supply current when the supply goes up.
Higher voltage can increase the 1/f noise in JFETs, so a higher voltage can also be bad.