SPARTANBURG, S.C.—Liars and lawsuits.

Both words figure prominently in the increasingly heated feud between Ted Cruz and Donald Trump, the two leading contenders in the Palmetto State primary.

Three days before South Carolina Republicans head to the polls, Cruz is challenging Trump to make good on his threat to sue him over an ad the Texas senator is running that hits Trump’s position on abortion. Cruz said he’d be happy to “take the deposition myself.”

Referring to Cruz, Trump said at a rally in the state this week: “I’ve never seen a human being lie so much.” In a statement Wednesday, Trump said he has grounds to sue over Cruz’s ad, which uses clips from a 1999 interview in which the real estate mogul describes himself as “very pro-choice.” And, in true Trump fashion, he went even further.

“Likewise, if I want to bring the lawsuit regarding Senator Cruz being a natural born Canadian, I will do so,” Trump said in a statement. “Time will tell, Teddy.”

The rivalry grows as a new NBC News/Wall Street Journal survey taken after last weekend’s debate here finds that nationally, Cruz is pulling ahead of Trump, 28 percent to 26 percent. Marco Rubio garners 17 percent. The RealClearPolitics polling average, however, has Trump still leading the field by 11 points nationally.

“There’s a new national frontrunner on the Republican side,” Cruz exclaimed while kicking off a rally here at a prep school gymnasium.

In South Carolina, Trump continues to lead the polls with 34 percent of the support, according to the RCP polling average. Cruz and Rubio, who received the vaunted endorsement of Gov. Nikki Haley on Wednesday, are neck and neck at roughly 17 percent.

South Carolina, with its conservative and evangelical bent, is a critical state for Cruz. The senator has staked his campaign on wooing and consolidating the party’s most conservative voters. At the end of one rally here, Cruz sounded more like a pastor than a politician. “Awaken the body of Christ, that we might pull back from the abyss,” he told audience members after asking them to pray for him.

Cruz is counting on a strong finish here to help propel him into Super Tuesday on March 1, when a slew of southern states — where his campaign is well positioned — host primaries.

That position also puts a large target on Cruz’s back here. The senator is fending off attacks this week from two angles. If Trump trounces him in South Carolina, or if Rubio beats him out for second place, Cruz would be bruised. Both Trump and Rubio have been calling Cruz a “liar,” arguing he has distorted their records for political gain and engaged in shady campaign tactics.

“He’s lying, and I think that’s disturbing,” Rubio said earlier this week.

Over the weekend, after “legal review,” South Carolina television stations pulled a super PAC ad supporting Cruz that attacked Rubio over immigration.

Trump has been going after Cruz since the Iowa caucuses, when the senator’s campaign told supporters Ben Carson was dropping out. Now, Trump’s campaign has issued Cruz a cease-and-desist letter over Cruz’s ad targeting Trump’s earlier abortion statements and is threatening a lawsuit. Cruz read the letter aloud during a press conference after a campaign stop in Seneca.

“You know I have to say to Mr. Trump, you have been threatening frivolous lawsuits for your entire adult life. Even in the annals of frivolous lawsuits, this takes the cake,” Cruz said, encouraging Trump to file the suit. “It is a remarkable contention that an ad that plays video of Donald Trump speaking on national television is somehow defamation.”

Cruz suggested he would delight in getting Trump to testify under oath. “I’ll point out, it didn’t work out very well for Bill Clinton,” he added. Cruz also pounced on Rubio, accusing him of “behaving like Donald Trump with a smile.”

But Cruz is doubling down on the ad targeting Trump. “The ad @realDonaldTrump wants me to pull,” he tweeted with a link to the spot. “We’re running it more frequently because you deserve to know the truth.”

The feud figures to sharpen over remaining days of the campaign here, and it could be something that overshadows Cruz. “If you’re defending the actions of your campaign and talking about lawsuits, that’s not your core message. That’s a negative,” said South Carolina GOP strategist Chip Felkel.

But South Carolina Rep. Jeff Duncan, who is campaigning for Cruz around the state, said Trump is running scared. “That seems like desperation now to drive a narrative. That signifies the momentum that Ted Cruz has, really,” he told RCP.

“This feels very similar to Iowa,” he said, referring to Cruz’s pre-caucus lag in the polls before winning there. “I think you’re going to see Ted Cruz as the leader of the pack at the end of the night.”

Cruz supporters say Trump’s threats only fuel their support for their candidate. “He’s too arrogant, and can’t produce,” said Mickey Dunagan of Boiling Springs. He came to see Cruz with his wife, Lucille, who made up her mind for Cruz after hearing him speak here. Dunagan described Rubio as a “car salesman.”

Elizabeth Carlo, a nurse from Greer, is still making up her mind between Cruz and Rubio. Cruz “is about protecting the Constitution and is very knowledgeable,” she said. “But Rubio … can take on Clinton. And I don’t want Clinton. That’s why I haven’t made up my mind.” Asked about Trump, she shook her head, calling him “dangerous.”

Bill Palmer, an undecided voter from Spartanburg, said he would probably make up his mind by Friday. He said he likes Cruz’s conservatism, likes Rubio, and has “a hard time swallowing Trump.”

“It’s a bit of a bloodbath,” he said of the political landscape here.

Rebecca Berg and Ellie Potter contributed to this report.