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First, the party would have to decide whether they would hold an open primary or a party convention.

A convention would likely favor a conservative candidate, but party members will probably want a candidate to be tested through the primary process.

Democrats are hoping that former Gov. Mark R. Warner will return to politics if John Warner, who is not related, retires.

The former governor told The Times late last month that he is weighing his options and will have a better idea of his future once the senator announces his decision in the fall.

“I think I’ve got a voice that I would like to get back engaged in public service, and whether that is the direction of the U.S. Senate or whether that is the direction of the governorship, I have yet to make a determination,” Mr. Warner told The Times.