“Democratic House members cannot be allowed to hide behind Speaker Nancy Pelosi when it comes to an impeachment inquiry of President Trump. They should – and must – vote to open an inquiry of impeachment so their constituents, country, and history can evaluate their actions,” Graham said in a statement.

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He added that there needs to be a “John Hancock moment,” referring to a signee of the Declaration of Independence, and that Democrats should “put their names on the line as supporting or opposing an impeachment inquiry.”

Pelosi already announced last week that the chamber would launch a formal impeachment inquiry over Trump's dealings with Ukraine, and she maintained during a press conference on Wednesday that a formal vote to begin the process wasn't necessary.

"There's no requirement that there be a floor vote. That is not anything that is excluded and, by the way, there are some Republicans that are very nervous about our bringing that bill — that vote to the floor," Pelosi told reporters Wednesday.

Still, Trump and some of his GOP allies are seeking to put pressure on Pelosi to hold a vote, viewing it as a potential political win for them because it would force Democrats in swing districts, and those who unseated Republicans during the 2018 midterms, to go on the record on the impeachment effort.

Axios reported on Thursday night that the White House is planning to send a letter to Pelosi as early as Friday saying Trump doesn't have to comply with demands from Democratic lawmakers until the House formally approves an impeachment inquiry.

Graham himself has previously dared House Democrats to bring the formal launch of an impeachment inquiry up for a vote.

"Those who believe that the transcript is a 'Smoking Gun' for impeachment, do something about it - have the courage of your convictions," he tweeted last month.