During his press conference, Graham used posters as visual aids to argue that the House’s impeachment inquiry into Trump deviates from the impeachment inquiry into President Bill Clinton. Graham's resolution has 44 Republican co-sponsors.

McConnell said Thursday that no time has been set yet for a vote but that he "obviously" supports the measure.

Senate Republicans have lambasted House Democrats for holding depositions behind closed doors and for declining to hold a vote that would formalize an impeachment inquiry, which happened in both the Clinton and Nixon impeachments but is not required by the constitution.

But Trump’s conservative allies in the House are growing frustrated that their Senate counterparts aren’t doing more to protect the president. Some House Republicans, who have been using their limited tools in the minority to disrupt Democrats’ impeachment probe, think the GOP-led Senate should use their powers to step up their defense of Trump.

Some are calling on Graham to hold his own impeachment hearings and haul in GOP witnesses as a way to counter-program the Democratic-led impeachment probe — and it’s unclear if Graham’s disapproval resolution will be enough to satisfy their demands.

“There are members of the House who would like [Graham] to take that to the next level,” Rep. Lee Zeldin (R-N.Y.), who has emerged as one of Trump’s most vocal defenders in the impeachment fight, told POLITICO on Wednesday. “The Senate Judiciary Committee might be that vehicle to let the American people more in on what the whole story is.”

But in the Senate, the minority on the Judiciary Committee can also bring their own witnesses. And other lawmakers point out that it makes sense that House Republicans are on the frontlines defending Trump since impeachment starts in the lower chamber. Plus, many senators say they need to stay neutral right now in case they wind up as jurors in an impeachment trial.

When asked about criticism from Trump allies that he needed to do more to counteract impeachment proceedings, Graham said there was only so much he could do and encouraged House Republicans to remain focused on the process.

“The Senate is different than the House,” Graham said. “I just don’t have the ability to call anybody I want to call, the other side gets a say. And I would be careful what you wish for.”

Earlier this week, Trump urged Republicans to “get tough and fight” for him on impeachment – a request House Republicans appear to be taking to heart. Dozens of GOP lawmakers stormed a secure facility Wednesday where impeachment investigators depose witnesses to protest the closed-door impeachment interviews – a stunt that caused a five-hour delay in testimony from Laura Cooper, a Pentagon official. The members refused to leave, which prompted intervention from the House sergeant-at-arms.

House Republicans also unanimously supported a failed effort to censure Rep. Adam Schiff (D-Calif.), who has emerged as Trump’s favorite punching bag in the impeachment fight.

And Trump is clearly taking notice: he fired off a tweet that specifically praised House Republicans, making no mention of the Senate GOP.

“Thank you to House Republicans for being tough, smart, and understanding in detail the greatest Witch Hunt in American History,” Trump tweeted Thursday morning. “It has been going on since long before I even got Elected (the Insurance Policy!). A total Scam!”

Jordan Sekulow, a member of Trump's legal team and son of the president’s lawyer Jay Sekulow, praised Graham for the resolution on his father's radio show this week.

"It sends a huge message to Democrats like Pelosi and Schiff and all those Democratic presidential candidates," he said. "Wherever you’re talking about impeachment, you’re going to fail. If you move that way, you’re going to fail.”

Graham, who had lunch with Trump and a group of senators Thursday, said the president still feels slighted by the impeachment inquiry.

“He would like the process to be exposed for being basically unfair,” Graham said. “He feels like it never stops. he feels like he doesn’t have a real fair chance of being president of the United States.”

Graham added that he spoke to acting chief of staff Mick Mulvaney and that the White House is working on getting a messaging team together to address impeachment. Graham later clarified in a tweet he "did not mean to leave some with the impression the White House needed to hire a new team to handle impeachment." He said the discussion was more about strategy, not personnel.

Senate Republicans insist that the Graham resolution is not about pressure from Trump on the GOP to do more on impeachment. Some senators, including Sens. Roy Blunt (R-Mo.) and Shelley Moore Capito (R-W.Va), said they had not heard anything from the president. They argue instead that it’s about fairness.

“It’s the process, it’s just unprecedented and it’s just unfair,” Capito said.

“Hopefully it gets the House to let them know that 'hey we’re watching on this side' if you’re going to conduct a hearing on that side with no due process,” added Sen. James Lankford (R-Okla.).

The Graham resolution could both please Trump and put pressure on Republicans like Sen. Mitt Romney of Utah, who have appeared open to considering the evidence in the House impeachment inquiry. Even Republicans who have declined to offer predictions on how the inquiry might end up have attacked Democrats for how they've handled impeachment.

But Romney appeared open Thursday to the Graham resolution.

“I’ll take a look at that, " Romney said. “I’d like to see a vote taken in the House to see if there’s support for an impeachment process. I’d love to see a more open process and I think it’s important we protect the whistleblower according to whistleblower law.”

Burgess Everett contributed to this report.

