Hurd agreed: “I … want to know who did Rudy Giuliani actually talk to. I think having Mayor Giuliani come and testify is important.”

The president’s lawyer also got some harsh words from retiring GOP Rep. Francis Rooney of Florida, who said recently he would be open to impeachment.

“I don’t buy having an amateur diplomat — quasi diplomat out there — disrupt the work of our paid professionals,” Rooney told Jake Tapper on CNN’s “State of the Union.”

Himes added that the investigation could still continue without Giuliani and said that Speaker Nancy Pelosi would like to wrap up the impeachment inquiry by the end of the year.

Democrats are hoping to tie Trump to Giuliani’s efforts to pressure Ukrainian officials to investigate Biden and his son Hunter. Although Hunter Biden sat on the board of Burisma, a Ukrainian energy company, while his father was vice president, there is no evidence that either of them did anything improper.

Last week, former and current administration officials voiced concern about Giuliani’s pressure campaign. And Bill Taylor, the top U.S. diplomat in Ukraine, is expected to testify Tuesday before congressional investigators. Rooney told CNN that Taylor’s interview would be key.

“What I’ve heard so far is quite troubling,” he said. “It’s quite troubling that we had a diplomatic outreach of civilians delegating our paid public servant diplomat.”

“I … want to know who did Rudy Giuliani actually talk to. I think having Mayor Giuliani come and testify is important," Rep. Will Hurd told CBS. | Chip Somodevilla/Getty Images

Gordon Sondland, President Donald Trump’s ambassador to the European Union, told House investigators last week that he did not agree with the president’s directive to contact the former New York mayor to discuss Ukraine. He added that Giuliani sought to tie a White House visit with Ukraine’s new president with requests that the Ukrainian government investigate the Biden family.

Trump’s former top Russia adviser, Fiona Hill, also testified last week that she raised concerns about Giuliani with White House officials, including then-White House national security adviser John Bolton.

Administration officials did not directly criticize Giuliani’s work on Sunday.

“John Bolton never complained to me about it, no one at NSC ever complained to me about anything that was going on,” Mick Mulvaney, the acting White House chief of staff, said in an interview with Fox News’ Chris Wallace.

Secretary of State Mike Pompeo also declined to criticize Giuliani’s efforts to dig up dirt on Biden, stating on ABC’s “This Week With George Stephanopoulos” that “private citizens often are part of executing American foreign policy.”

Pompeo acknowledged that Giuliani provided him with a dossier of information related to the Ukraine, but said that he never reviewed the information. Pompeo further defended the administration’s handling of the removal of Maria Yovanovitch as ambassador to Ukraine after Trump allies accused her of being disloyal. Yovanovitch told Congress she was removed over “unfounded and false claims.”

“She still works at the State Department, she’s a foreign service officer in good standing,” Pompeo said. “You know this, George. Ambassadors serve at the pleasure of the president, and when a president loses confidence in an ambassador, it’s not in that ambassador, the State Department or America’s best interest for them to continue to stay in their post.”

Mike McKinley, a former adviser to Pompeo, told congressional investigators that he resigned over Trump’s attacks on Yovanovitch, and reportedly urged Pompeo to issue a statement about Yovanovitch. But Pompeo said Sunday he received no such request.

“Not once … did Ambassador McKinley say something to me during that entire time period,” Pompeo said.

Republicans also reiterated their criticism of the way Democrats are conducting the impeachment inquiry. Pompeo said it was “deeply unfair” that State Department lawyers were not allowed for officials’ closed-door testimony, while Hurd said that Republicans on the committee were not getting enough information after the depositions and that minority staff were not given the same level of access to the meetings as Democratic staff.

“My concern is with the information the committee has access to after these depositions are done,” Hurd said. “The evidence that is given in those meetings has not been made fully available to Republicans.”

Hurd further said that if there was evidence of a quid pro quo between the White House and Ukraine, he wanted to know what it was for.

“Was it for the investigation of the previous election? Or was it to get dirt … for 2020?” he said. “We haven’t heard yet. We haven’t heard from any Ukrainian official that felt like there was this arrangement.”

But Himes said that Republicans were focusing on the impeachment process to avoid discussing Trump’s actions. He added that eventually the information in closed-door hearings would become public.

“An investigation doesn’t happen in the light of day, but I will tell you that there will be open hearings,” Himes said. “This will all ultimately be all out there for the American people to see.”