Graham referred to a press conference last week by acting White House chief of staff, which sparked a days-long clean up after Mulvaney said security aid to Ukraine was dependent partly on the country investigating a conspiracy theory undermining the conclusion that Russia interfered in the 2016 election.

"What's missing here I think is that coordinated effort to put somebody in charge of developing a message and delivering it. I believe that's about to be corrected, I hope," Graham said.

Sen.(R-S.C.) said on Wednesday that he thought the White House needs to do a better job on impeachment messaging, and urged them to take a page out of former President Clinton's book.

"I like Mulvaney but the news conference is not exactly what you want. You want people who understand the legal implications of what you say, as well as the political implications," Graham said.

Trump has faced pressure from current and former administration officials to hire a new chief strategist amid high-profile missteps as they respond to the House Democrats' impeachment inquiry, which is centered on Trump asking Ukraine to investigate the Bidens and if he tied aid to the country opening a probe.

The role of White House chief strategist has been vacant since Stephen Bannon was forced out of the position more than two years ago. Trump has also refused to start an impeachment "war room."