The mailers urge voters to back a Libertarian or Independent voter over Mr. Hawley and do not include required disclaimers identifying who paid for them.

“Voters telling me about fake & illegal mailers pro-Dem group is putting out. There should be an FBI investigation,” Mr. Hawley, the state attorney general, wrote on Twitter.

He called for Democrat incumbent Sen. Claire McCaskill to cooperate in a probe.

“As the beneficiary of this Kavanaugh-style illegal smear campaign, I hope @clairecmc will cooperate with the investigation fully #MoSen,” tweeted Mr. Hawley.