Kansas and Arizona say yes. NRO has a detailed look at the case:
A major voting-law case that the Supreme Court looks likely to take next term, Kobach v. Election Assistance Commission, sets out a dispute between states and the federal government over whether Kansas can require inclusion of a proof-of-citizenship requirement on the National Mail Voter Registration Form. As the name suggests, this form is a single document that can be used nationwide to register to vote, with varying instructions for each state; the Election Assistance Commission (EAC) is a federal agency whose responsibilities include creating and administering this form…
As Kobach states in petitions for Kansas and co-plaintiff Arizona, maintaining confidence in our voter-registration system — in this case, by removing ineligible voters from the voter rolls — is central to the National Voter Registration Act, the 1993 law that created the standard federal registration form. But it’s the law’s other provisions, on expanding ballot access, that seem to get all the attention, and the EAC barely mentioned the NVRA’s election-integrity goals in its 40-page decision denying Kobach’s request.
Kobach argues Supreme Court guidance is needed because the case is of paramount national importance. “It’s a really profoundly important case,” Kobach said. “The founding fathers were emphatic that the states get to decide who is a qualified voter and who is not. It was a critical point in the Constitution that the federal government would have to follow the states on this matter.”