Senate Judiciary Committee Chairman Lindsey Graham said Sunday that if House Democrats refuse to bring in the “whistleblower” during their impeachment hearings, “this thing is dead on arrival in the Senate.”

“It’s impossible to bring this case forward in my view fairly without us knowing who the whistleblower is and having a chance to cross-examine them,” said Mr. Graham on Fox’s “Sunday Morning Futures.” “If they don’t call the whistleblower in the House, this thing is dead on arrival in the Senate.”

House Intelligence Committee chairman Adam Schiff rejected Saturday two names on the GOP’s list of eight requested witnesses — the “whistleblower” and Hunter Biden — saying he refused to turn the panel’s hearings this week into a “sham investigations into the Bidens.”

Mr. Schiff also said that any testimony offered by the anonymous “whistleblower” who filed the original complaint on President Trump’s July 25 phone call with the Ukraine president would be “redundant and unnecessary,” given that the information was secondhand.

A public appearance would put the whistleblower’s “personal safety at grave risk,” said Mr. Schiff in his letter to ranking member Devin Nunes, who sent the chairman a list of eight names and asked for them to be called to testify at the public impeachment hearings, which begin Wednesday.

Mr. Graham accused House Democrats of trying to hide the whistleblower’s reported connections to two staunch Trump critics — former Vice President Joseph R. Biden and former CIA director John Brennan — as well as his communications with Mr. Schiff’s staff before the complaint was filed.

Several media outlets have named the “whistleblower” since Real Clear Investigations revealed the person’s alleged identity on Oct. 30, while Facebook said Friday that it would scrub all mentions of the individual’s name on its platform, citing its “coordinating harm policy.”

“The whistleblower is foundational to what they’re doing in the House, and the fact that they don’t want to call him tells you everything you need to know about how valid this effort is to impeach the president,” Mr. Graham said. “It is impossible to conduct an impeachment inquiry when the chief complaining witness is unknown to the president, not subject to cross-examination.”

The South Carolina Republican alleged that “this is not about Schiff finding the truth, this is about Schiff trying to destroy the Trump presidency.”

“It would blow them out of the water if in fact the whistleblower was connected to the Democratic candidate and came from the CIA world that’s been trying to destroy the Trump presidency before he got elected,” Mr. Graham said. “That’s why they don’t want you to know who this person is.”

Sen. Amy Klobuchar, Minnesota Democrat, argued that the whistleblower’s testimony was unnecessary, given that other higher-ranking government officials have testified in closed hearings about the phone call.

“[A]ll sources tell us that what the whistleblower said was secondhand information, and we now have firsthand information of diplomats, military people, others that actually heard these actions occur where the president was pushing for Ukraine to start an investigation against his political opponents in exchange for military assistance,” said Ms. Klobuchar on CNN’s “State of the Union.”

Mr. Trump has denied any quid pro quo, while Republicans have sought information on Hunter Biden’s seat on the board of Burisma, a gas company under investigation by the government and Joe Biden’s push for the firing of a Ukrainian prosecutor.

“I think it’s important that we find out what really did happened regarding the firing of the prosecutor and what was the conflict of interest that Hunter Biden had,” said Mr. Graham.

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