The Senate Judiciary Committee on Thursday held off on approving subpoenas for documents and testimony from top Obama administration officials as part of the panel's investigation into the origins of the Russia probe during the 2016 presidential election.

The committee on Thursday was slated to vote on whether to give Chairman Lindsey Graham, R-S.C., the power to issue subpoenas, which would cover documents, communications and witness testimony in a public setting or behind closed doors for any “current or former executive branch official or employee involved in the ‘Crossfire Hurricane’ investigation.”

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But after two hours of debate, Graham decided to hold off on issuing the more than 50 subpoenas to former Obama officials until the panel convenes again next week on June 11.

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Under committee rules, Graham cannot unilaterally issue a subpoena. The committee chairman can only issue a subpoena with the consent of the ranking member, Sen. Dianne Feinstein, D-Calif., or by a committee vote.

Feinstein, during the business meeting to consider the subpoenas Thursday, slammed Graham's proposal to authorize the subpoenas, saying if approved, it would give him "unbridled authority to go after Obama era officials," something, she said, she could not support.

Feinstein also noted that the last subpoena issued by the Senate Judiciary was in 2017, when she and then-Chairman Chuck Grassley, R-Iowa, agreed to subpoena former Trump campaign chairman Paul Manafort.

"I can’t accept this dragnet authority to conduct politically motivated investigations," Feinstein said Thursday.

Feinstein also said that if the committee is going to engage in a deeper investigation into the origins of the Russia probe, that Democrats would like former Special Counsel Robert Mueller to appear for testimony before the committee -- a request she and Democrats on the committee made to Graham last year.

Graham agreed to Feinstein's request on Thursday.

"I think we need to understand how the Mueller team worked and I am not opposed to having someone from the Mueller team come over here," Graham said Thursday in response to Feinstein's request.

It is unclear whether anybody from Mueller's team would be subpoenaed to appear for testimony, or would appear voluntarily.

Graham and Republicans on the committee are seeking testimony from former FBI Director James Comey, former FBI Deputy Director Andrew McCabe, former Director of National Intelligence James Clapper, former CIA Director John Brennan, former Deputy Attorney General Sally Yates and others.

The subpoena authorizations would also cover any documents, communications and testimony “related to any aforementioned matter” from current and former officials, including Trisha Anderson, James Baker, Dana Boenta, Mary McCord, Andrew McCabe, Loretta Lynch, Jonathan Moffa, Bruce Ohr, Lisa Page, Peter Strzok, Joseph Pientka, John Podesta, Samantha Power, Susan Rice, Rod Rosenstein, Bill Priestap, and Sally Yates among others.

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The authorizations would also cover individuals involved in the Steele Dossier, including Fusion GPS founder Glenn Simpson and Nellie Ohr.

Graham announced last month that his investigation would specifically focus on unmasking, abuses of the Foreign Intelligence Surveillance Act, and the appointment Mueller as special counsel in May 2017.

But Sen. Sheldon Whitehouse, D-R.I., argued Thursday that Republicans are only focused on FISA abuse, while there have been other "issues" with the Justice Department, which he said had not received proper inquiries, taking issue with the FBI's "investigation" into allegations leveled against then-Supreme Court nominee Brett Kavanaugh in 2018.

“There is something a little bit questionable to me, a red flag fluttering, when I see this as the only exercise in committee oversight," Whitehouse said.

But Sen. John Kennedy, R-La., maintained that some in "Washington" engaged in FISA abuse.

“Power corrupts," he said. "And absolute power is kind of cool.”

And Graham weighed in on the back-and-forth, saying: "I think there’s a really good chance that people could go to jail for abusing the FISA process."

Sen. Ben Sasse, R-Neb., hit back at Graham, urging him to authorize the subpoenas as planned, and accused Democrats of "trolling for soundbites."

Nevertheless, Graham said he would allow Democrats to have their say.

Next, Sen. Kamala Harris, D-Calif., weighed in, arguing that the committee's focus on subpoenas in their Russia investigation shows that "it does not care" about civil unrest and police violence.

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Harris moved to offer an amendment to prevent the issuing of subpoenas until the Judiciary Committee holds an investigation into potential orders given by Attorney General Bill Barr earlier this week in disbursing protesters from Lafayette Park in front of the White House so that President Trump could visit St. John's Episcopal Church. Trump visited the church after protesters set the church to fire late Sunday night amid a demonstration that turned violent in response to the death of Minneapolis man George Floyd in police custody last week.

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Graham told Harris that the committee would consider her amendment, and said he believed the "subpoena debate" is "important to carry over" to next week.

Meanwhile, the Senate Homeland Security and Governmental Affairs Committee also voted to authorize subpoenas Thursday for records and testimony from government agencies and individuals related to the Russia probe, the Justice Department inspector general's review of that investigation, and the "unmasking" of U.S. persons affiliated with the Trump campaign, transition teams, and the Trump administration.

The subpoena authorization comes a day after former Deputy Attorney General Rod Rosenstein testified before the committee, saying that he would not have signed a FISA warrant renewal for former Trump campaign aide Carter Page had he known about the since-revealed misconduct surrounding those warrants -- while faulting the FBI for its handling of the documents.

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Graham plans to hold “multiple, in-depth congressional hearings regarding all things related to Crossfire Hurricane” beginning this month.

Crossfire Hurricane was the FBI’s internal code name for the investigation into whether members of the Trump campaign were coordinating with Russia during the 2016 presidential election. Mueller's team eventually announced that it found no evidence of coordination.

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The first phase of the panel’s investigation “will deal with the government’s decision to dismiss” the case against former national security adviser Michael Flynn, as well as “an in-depth analysis of the unmasking requests made by Obama Administration officials against Gen. Flynn.”

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Meanwhile, Graham said the second area of inquiry for the committee would take place later this summer and would be focused on FISA abuses outlined by Justice Department Inspector General Michael Horowitz. Graham said that part of his investigation would focus on the FISA warrants obtained against Page.

Graham added that the committee will also look at "whether Robert Mueller should have ever been appointed as special counsel.”

Fox News' Chad Pergram contributed to this report.