Trump has repeatedly called on Barr to investigate how the FBI began its Russia probe. John Durham, the U.S. attorney for Connecticut, is leading that effort. The Justice Department’s Office of the Inspector General has also been conducting an investigation into aspects of the FBI's Russia probe. That report is expected to be released in the near future.

Senate Democrats on the Judiciary Committee also sent a letter to Barr on Thursday that questioned the attorney general's ability to be impartial in any matter related to Trump and Ukraine. In their letter, the senators cited Trump's July 25 call with President Volodymyr Zelensky of Ukraine, in which he asked Zelensky to investigate the Biden family. During that call, Trump said that both Barr and his personal attorney Rudy Giuiliani would get in touch with the Ukrainian president.

"This raises legitimate questions about your knowledge of the activities of Mr. Giuliani and others, as well as the actions that you have taken and your discussions with the President and White House about these investigations," the senators wrote.

The Democratic senators also cited Barr's visit to Italy to look into the origins of the Russia investigation, and noted that Trump told Zelensky he would "like to have the Attorney General call" to discuss a conspiracy theory related to Ukraine and the 2016 election.

"Impartial enforcement of the law is essential to give the American public confidence in the Justice Department's work," the senators wrote. "Your personal connection to these matters creates the appearance of a conflict of interest and gives rise to questions about whether the Department is being used to advance the President's personal interests."

But Republicans defended the Justice Department’s decision to classify its review as a criminal investigation. Sen. Lindsey Graham (R-S.C.) further predicted on Fox News’ “Sunday Morning Futures with Maria Bartiromo” that the Justice Department’s inspector general report would be damning. He also touted his resolution, introduced Thursday, that condemns the House impeachment inquiry.

“Fifty Republicans have told the Nancy Pelosi House Democrats that what you're doing to the president is unfair,” Graham said, citing the number of co-sponsors to his resolution.

Schiff was unfazed by Graham‘s resolution. He instead touted a Friday court decision in which a federal judge ruled that the Justice Department needed to turn over former special counsel Robert Mueller’s grand jury evidence to the House Judiciary Committee.

The decision came after House Democrats secured damning testimony last week from William Taylor, the United States’ top diplomat in Ukraine, who told investigators that Trump wanted to make military aid and a White House meeting with the Ukrainian president conditional upon a Ukrainian government investigation into his political rivals.

“I think what we’re seeing really is Lindsey Graham trying to respond to the devastating testimony that’s come out from Ambassador Taylor and others,” Schiff said on “This Week."

Schiff said the House Intelligence Committee would continue moving forward, and described former national security adviser John Bolton as “an important witness.” He reiterated that the House would have public hearings but declined to say when.

“I don’t want to give precise timing because ,in part, we’re struggling with the White House’s continuing efforts to obstruct our investigation, to obstruct witnesses coming in,” Schiff said.