Cuban certainly can try. The House website says: "Although the Constitution does not require the Speaker to be a Member of the House, all Speakers have been Members."

The billionaire impresario said in a tweet he may run for the House's top spot.

Cuban floated the idea after the front-runner, House Majority Leader Kevin McCarthy, suddenly withdrew from the race Thursday, leaving two members officially running: Reps. Daniel Webster of Florida and Jason Chaffetz of Utah.

Webster announced his candidacy in a video on Monday. "As speaker of the House, I would commit to serving you by promoting principle over power," he said.

Chaffetz announced his bid on Sunday in a statement. "I came to Congress to help fix problems, and as speaker I will fight every day to make that happen," he said.

On Friday, Rep. Darrell Issa of California told CNBC's "Squawk Box" he may run.

Many observers have also mentioned Rep. Paul Ryan for the post, but he said he's not interested. Rep. John Boehner, whose resignation plans created the vacancy, said Thursday he would stay on until a successor is selected.

In addition to duties leading the House, the speaker is second in line for the president, after the vice president.