Originally Published: 24 AUG 22 10:31 ET

By Tierney Sneed, CNN

(CNN) -- Sen. Lindsey Graham on Wednesday told a federal court that it should bar a grand jury's questioning of the South Carolina Republican "on all the topics" sought by Fulton County District Attorney Fani Willis in the subpoena for Graham's testimony as part of an investigation into plots to illegally influence Georgia's 2020 election results.

Graham also asked the court to block questioning of him on "other topics" suggested by outside groups that have weighed in on the dispute over the subpoena in friend-of-the court briefs.

The litigation over the subpoena is back before US District Judge Leigh Martin May of the Northern District of Georgia after an appeals court on Sunday paused her ruling requiring that Graham appear before the grand jury.

May had previously rejected Graham's request that she quash the subpoena outright. The appeals court has since instructed her to consider whether the subpoena should be partially quashed or modified in accordance with the Constitution's Speech or Debate Clause, which shields lawmakers from some law enforcement actions when it covers certain conduct that is part of lawmakers' legislative duties.

Graham, however, did not scale back his arguments in his latest brief for why he thinks the entire scope of the subpoena should be seen as off limits under the Constitution. He argued Wednesday that Willis "should bear the burden of identifying the questions she wants to ask or, at a minimum, the specific topics she wants to cover," and said that "if there is debate about whether those topics are permissible, she must offer evidence that those topics are not protected by the Clause."

"To hold otherwise is to allow the District Attorney to evade the Speech or Debate Clause by requesting amorphous testimony and then, like a moving target, springing new or elaborated questions or issues on the Senator; the District Attorney, after all, is in the best position to know what she wants to ask (assuming this is not a disorganized fishing expedition)," Graham said.

"For that reason, if anyone suggests a further line of inquiry, Senator Graham will address it in his Supplemental Reply Brief."

Graham said that the conduct Willis has signaled she is interested in questioning -- including phone calls Graham had with Georgia election officials about the state's review of the ballots cast in the 2020 presidential election -- were part of an informal investigation he was conducting as part of his duties as a US senator.

The South Carolina Republican pointed to several examples of legislative activity that he said was connected to the conduct in Georgia that the District Attorney now wants to question him about. He cited research he was doing before voting to certify Joe Biden's election win, legislation he has co-sponsored to amend the Electoral Count Act and elections-related hearings he presided over when he chaired the Senate Judiciary Committee.

"Because all the District Attorney has in fact sought are materials central to this protected investigation and the motives behind it, complete quashal remains appropriate," Graham's brief said. To the extent the court disagrees, he wrote, the court "should at least partially quash or modify the Subpoena to prohibit any questioning about the investigation and the motives behind it."

Willis' office has until next Monday to respond and then a reply from Graham is due by August 31