The change was reportedly prompted by officials after Mr Harrison declined to participate unless the Republican incumbent took a coronavirus test, which he reportedly declined.
Both candidates will participate in separate 30-minute interviews hosted by television station WSPA on Friday.
In the hours leading up to the event, Mr Harrison called on debate moderators and Senator Graham to take Covid-19 tests.
Senator Graham, suggesting that Mr Harrison sought “special treatment” to get out of the debate, said that he would “follow the guidance of my doctors, not my political opponent.”
“I have taken the coronavirus threat to our nation and state seriously,” he claimed in a statement on Thursday. “I have acted responsibly, seeking testing when required and following the advice of physicians.”
The senator, who chairs the Senate Judiciary Committee, has said that confirmation hearings for Donald Trump’s Supreme Court nominee Amy Coney Barrett will move forward within the coming days, following the president’s announcement at the White House Rose Garden, after which at least a dozen attendees tested positive for the disease – including the president.
On Friday, the senator suggested Mr Harrison was “skipping tonight’s debate” while relying on a political stunt to avoid questions about his platform.
“South Carolinians do not appreciate Harrison putting himself above others,” he said on Twitter. “If Mr Harrison is not able to interact with South Carolinians on the same terms they live their lives, he should not be their senator.”
“No such luck, Lindsey. I’ll be there,” Mr Harrison replied. “Thank you to our hosts for accommodating Senator Graham’s refusal to take a Covid test, and changing the debate format to keep everyone safe.”
The back and forth has underscored the outrage aimed at Republican officials for failing to take seriously a public health crisis that has infected millions of Americans, while the president has downplayed the gravity of the illness and the White House has refused to provide public details about his testing and the contact tracing in place to track people close to him.
Mr Harrison said he would “not allow politics to put my family, my campaign staff, Senator Graham’s staff, and members of the media at unnecessary risk” by moving forward with a debate that could expose attendees to the virus – a week after the president was hospitalised days after appearing on the debate stage next to Joe Biden.
During Wednesday night’s vice presidential debate, Mike Pence and Kamala Harris sat 12-feet apart, guarded by plexiglass dividers.
Mr Harrison brought his own plexiglass shield to his debate with the senator on 3 October.
Recent polling shows the candidates locked in a statistical tie, with more recent polls showing Mr Harrison in a slight lead, in a Republican stronghold.
Senator Graham is seeking a fourth term; Mr Harrison is the first black chair of the South Carolina Democratic Party and associate chairman of the Democratic National Committee.
A Quinnipiac University poll from the end of September shows each candidate receiving 48 per cent of the vote.
The nonpartisan Cook Political Report considers the race a “toss up.”
Senator Graham emerged as an early opponent of Mr Trump’s 2016 campaign but has become a reliable source of support in the GOP-controlled Senate.
In 2016, joining a Republican effort to block then-president Barack Obama’s Supreme Court nominee, the senator pledged he would not do the same if there was a vacancy on the high court under a Republican administration.
“I want you to use my words against me,” he said at the time. “If there’s a Republican president in 2016 and a vacancy occurs in the last year of the first term, you can say, ‘Lindsey Graham said, “Let’s let the next president, whoever it might be, make that nomination.”’”
However, following the death of Ruth Bader Ginsburg within weeks before Election Day, he has joined Republicans to press forward with the president’s nominee.