Mr. Blankenship did not provide evidence or a legal argument to support the claim. But scholars who have studied the law said they could see a reasonable argument to overcome it.

For Democrats, the prospect of a third-party entry in West Virginia is a gift. Democrats have been trying to foment Republican divisions, especially in races against sitting Democratic senators in states that President Trump won.

In Ohio, they have highlighted misgivings that another losing primary candidate, Mike Gibbons, has with the Republican winner, Representative James B. Renacci, who will face Senator Sherrod Brown in November. In Wisconsin, they have egged on the Republican combatants slugging it out for the right to challenge Senator Tammy Baldwin in the fall. And intraparty fighting in Arizona could imperil Republicans’ effort to maintain the seat being vacated by Senator Jeff Flake.

Mr. Blankenship would have little chance of winning the Senate seat outright, but if he could make it onto the ballot, he could siphon off support for West Virginia’s attorney general, Patrick Morrisey, who won the Republican nomination. That could propel Mr. Manchin to re-election.

And even if his legal challenge fails, Democrats said on Monday that Mr. Blankenship could torment Mr. Morrisey with negative ads, and reach deep into his pockets to demonize the Republican leadership in Washington.