By Clare Foran, CNN

(CNN) -- President Donald Trump's baseless claims of a rigged election loomed over a Georgia Senate debate on Sunday evening, in which Republican Sen. Kelly Loeffler and her Democratic opponent the Rev. Raphael Warnock sharply attacked each other ahead of a pivotal runoff election next month.

Loeffler did not explicitly say that she believes the presidential election was rigged -- as the President has falsely claimed -- when pressed during the debate, but did say "it's very clear that there were issues in this election," while Warnock criticized the Republican senator over her rhetoric, saying that she "continues to cast doubt on an American democratic election."

During the debate, Loeffler said Trump has "every right to every legal recourse," when asked if she stands by the President's baseless narrative about the election, but then attempted to turn attention to her own Senate race.

"The President was also clear that Georgians need to come out and vote for David Perdue and myself because of what's at stake," she said, referencing Trump's campaign rally Saturday night in the state stumping for both Republicans in the runoff.

An intense national spotlight is focused on the race and the stakes are high since its outcome, along with the result of a second Georgia runoff in January between Perdue and Democratic challenger Jon Ossoff, will determine control of the Senate in the new Congress.

If either Republican incumbent holds onto their seat, the GOP will be poised to maintain its Senate majority. But if both Democrats win, it would bring the balance of power to 50-50 in the upper chamber with Vice President-elect Kamala Harris able to step in and cast tie breaking votes.

The two Senate races advanced to a January runoff after none of the candidates cleared a 50% vote threshold in November to win outright.

The Senate races, however, have been overshadowed at times by Georgia's presidential election results, where President-elect Joe Biden turned the state blue for the first time in 28 years. Trump has blamelessly alleged voter fraud in the state and attacked statewide officials including Republicans Gov. Brian Kemp and secretary of state Brad Raffensperger.

Candidates sharply criticize each other

During the debate, Loeffler and Warnock spent much of their time attacking their opponent's character.

Loeffler repeatedly referred to her Democratic opponent as a "radical liberal" -- seeking to portray Warnock as she has throughout the campaign so far as extremely far left.

"Listen, I believe in our free enterprise system," Warnock said during the debate after Loeffler asked if he would renounce socialism and Marxism.

Warnock, for his part, targeted the GOP senator over stock transactions that have been a subject of intense scrutiny for their timing related to the coronavirus pandemic.

Addressing Loeffler, he said, "You dumped millions of dollars of stock in order to protect your own investments and then weeks later when there came an opportunity to give ordinary Georgians an extra $600 of relief, you said you saw no need and called it counterproductive."

Loeffler fired back, "I've been completely exonerated. Those are lies perpetrated by the left-wing media and Democrats to distract from their radical agenda. Since I got to the Senate, I've worked hard to deliver relief to Georgians during this pandemic and I'm continuing to do that."

Runoffs put Georgia in the spotlight

On Saturday night, Trump stumped for Loeffler and Perdue at a Valdosta, Georgia, rally, but once again falsely claimed he won the state and warned without evidence that the runoffs in January could be rigged. CNN has previously reported that Republicans were concerned that Trump could depress turnout among his base if he continued to rail against Georgia's election system. At one point, Trump welcomed Loeffler and Perdue to the stage for very brief remarks at the rally, but both senators were immediately interrupted with chants of "Stop the Steal" and "Fight for Trump."

Loeffler and Warnock confirmed their participation in Sunday's debate, while Ossoff confirmed he would participate in an earlier debate the same day, but Perdue declined, according to the Atlanta Press Club.

Ossoff still appeared before the press club earlier Sunday evening, while Perdue was represented by an empty podium. Ossoff fielded questions about his response to the coronavirus pandemic and used the platform to call out Perdue's absence.

"The reason that our country has lagged the entire world at the efficacy of our response to this virus, the reason that we are losing thousands of people per day to this virus is because of the arrogance of politicians like David Perdue," Ossoff said Sunday night. "So arrogant that he disregarded public health expertise, and so arrogant that he's not with us here today to answer questions."

Loeffler has unleashed an onslaught against Warnock, trying to portray her Democratic opponent and the 15-year leader of Martin Luther King Jr.'s historic church in Atlanta as an anti-police Marxist who will destroy America.

Warnock, in turn, has said that Loeffler wants to divide Georgia, and distract from her opposition to the Affordable Care Act and the health care insurance it provides millions of people in the middle of a pandemic.

At the age of 35, Warnock was chosen in 2005 to lead Ebenezer Baptist Church, and has since taken on issues in Georgia like overhauling the criminal justice code, and expanding voter registration and Medicaid.

Loeffler has described the work ethic she learned on her family's Illinois farm, becoming the first in her family to graduate from college and her work for the Intercontinental Exchange, the commodities and financial exchange company.

She then married Jeffrey Sprecher, the chairman of the New York Stock Exchange and bought a co-ownership in the WNBA's Atlanta Dream. At the end of 2019, Republican Gov. Brian Kemp appointed Loeffler to fill the seat left by the retiring Sen. Johnny Isakson.

This story has been updated with additional developments Sunday.