President-elect Donald J. Trump is well into filling out his cabinet and picking key advisers. But a move to challenge the vote tallies in three swing states — an extreme long shot to reverse his Electoral College majority — is advancing in the background, creating a noisy distraction on Mr. Trump’s Twitter feed and raising last-ditch hopes of some Hillary Clinton supporters.

In Wisconsin, elections officials said on Monday that a recount of the state’s nearly three million votes would most likely begin on Thursday. In Pennsylvania, Jill Stein, the Green Party presidential candidate, who initiated the recounts, filed a legal challenge of the results in state court. And the Stein campaign said it planned to request a recount in Michigan on Wednesday.

Neither Ms. Stein nor the Clinton campaign has found evidence of election tampering in any of the three states, where Mr. Trump beat Mrs. Clinton by a combined margin of only about 100,000 votes. But once Ms. Stein seized on the issue last week, the Clinton campaign said it, too, would join in the efforts to seek recounts.

Mrs. Clinton has been urged by her supporters to challenge the results, but she has done so only in a passive way. A campaign lawyer said that her team would “participate” in a manner that ensured fairness, but that it had “not uncovered any actionable evidence” of hacking or other interference.