Republican presidential candidate Ted Cruz speaks to supporters during a rally at the Durango Hills Community Center on Monday in Las Vegas. (Ricky Carioti/The Washington Post)

Ted Cruz’s presidential bid is in turmoil after repeated allegations of unsavory campaign tactics by his Republican rivals, leading some key supporters to call for a shake-up in the candidate’s message and strategy a week ahead of the crucial Super Tuesday primaries.

Aides and allies of the insurgent senator from Texas acknowledged in interviews this week that the campaign has been damaged by attacks on Cruz’s integrity from Donald Trump and Marco Rubio. They have pointed to a series of questionable tactics by the Cruz camp, including calls to voters suggesting that candidate Ben Carson was dropping out and the sharing of an inaccurate video suggesting Rubio had disparaged the Bible.

The video flap prompted Cruz to abruptly fire his chief spokesman, Rick Tyler, who posted the clip on social media, on Monday in an attempt to put his candidacy back on course. But the troubles continued Tuesday, when the campaign halted the sale of merchandise by a street artist whose social-media accounts include controversial and sometimes racist messages.

Cruz also weathered another wave of attacks from Trump on the eve of the Nevada caucuses. Trump told a rally in Sparks, Nev., that Cruz is “like a little baby — soft, weak little baby. . . . But for lying, he’s the best I’ve ever seen.”

Cruz and his aides say the accusations of deception are simply false. But with the issue dominating media coverage for more than a week, aides and supporters now acknowledge that the attacks have started taking a toll.

Louie Hunter, Cruz’s Georgia co-chair, said the allegations of untruths being pushed by Trump and Rubio have made their way to voters.

“I think both the Trump and the Rubio campaigns have seized on the narrative that if they say ‘liar’ enough, enough people are going to believe it,” Hunter said. “I think that has manifested itself into some people questioning, albeit incorrectly, the real moral character of Senator Cruz and of this campaign.”

The tumult comes at a crucial time for Cruz, who won the Iowa caucuses at the start of the month but finished third in both New Hampshire and, more disappointingly, South Carolina — a state filled with the evangelical Christians and tea party conservatives who make up his political base. After Tuesday night’s Nevada caucuses — which Trump won — Cruz is looking ahead to March 1, known as “Super Tuesday,” which features 11 state contests mostly in the South, including Cruz’s home state of Texas.

Cruz’s campaign has been focused on Super Tuesday since its inception, with the candidate calling the South a “firewall” that could help him gobble up delegates and secure the nomination. The campaign has poured time and resources into the delegate-rich region, believing its high percentage of religious and conservative voters would be natural Cruz supporters.

But with the Republican campaign in a new national phase, a Cruz aide acknowledged that the shouts of “liar” could have far-reaching implications for a can­didate who touts his slogan ­“TrusTed.”

For some supporters, some of the alleged dirty tricks — including a photo on a website created by the Cruz campaign that digitally added Rubio’s head onto the body of a man shaking hands with President Obama — were too much.

“I think when it looks like there’s a pattern of a Photoshopped picture or now a false tweet, that’s not good. It’s going too far, it’s crossing the line,” said Mica Mosbacher, a Cruz donor from Houston. “It was becoming a pattern. . . . When you see one Photoshop and now a false tweet, that was a pattern that needed to be stopped in its tracks.”

Cruz’s campaign did not respond to requests for public comment.

Some are questioning whether Cruz’s overall strategy of focusing on evangelical, conservative and, to a lesser extent, libertarian-leaning voters is too narrow.

The campaign needs to reach beyond its base of evangelical support, said Mike Gonzalez, a South Carolina pastor who was a co-chair for Cruz’s South Carolina evangelical efforts. “Reach libertarians, you know, moderates as well, and even the establishment people.”

Gonzalez and others think Cruz needs to focus more on his story and issues such as the Supreme Court. And, they say, focus less on Rubio and more on Trump. Cruz has repeatedly hit Rubio on immigration and foreign policy.

“In my opinion it’s a two-person race” between Cruz and Trump, said Bob Vander Plaats, an influential Iowa evangelical leader and a national co-chair for Cruz. “I would stay focused on your main competitor. I wouldn’t wage a two-front battle.”

Some advice to Cruz from top surrogates has been very public and unsolicited. David Limbaugh, a conservative commentator and Cruz supporter, wrote that Cruz needs to adjust his campaign by not focusing so much on religion, to quit parsing his opponents’ positions like a lawyer and to stop getting bogged down in arguments over the 2013 push for comprehensive immigration reform.

Steve Deace, a high-profile conservative Christian Iowa radio host in Iowa, penned a piece in Conservative Review urging Cruz to reboot his campaign.

“Stop telling us how much you like Trump personally. Stop saying things like ‘everyone on this debate stage would be better than Hillary,’ ” Deace wrote, referring to some of Cruz’s earlier overtures to the real-estate mogul. “Trump is playing for first place or go home. Now go and do likewise.”

Jason Schultz, an Iowa state senator who supported Cruz in the caucuses there, said the campaign has faced a difficult balancing act: If Cruz doesn’t respond to the allegations, he looks as though he is conceding that they are true; if he does, he takes away from his message.

“I get the sense that he left his message because he did try to respond and maybe spent too much time responding instead of talking about what Senator Cruz brings to the campaign,” Schultz said. “I don’t know if it’s ever helpful to get down into the gutter with the gutter fighters.”

Regardless, the attacks have stuck with voters.

Lauriel Bradley, a 29-year-old Republican from Clifton, Va., said she’s consistently been hearing two things about Cruz lately: “That he’s a liar and dirty.” Where has she been hearing this?

“People, the news, Facebook,” she said, sipping a latte in a coffee shop in Clifton. Bradley doesn’t plan to vote for Cruz on Super Tuesday, but she declined to say which candidate she plans to support.

Andrea Vaden, a Rubio supporter from Las Vegas who attended a Rubio rally there on Tuesday morning, said she learned about Tyler parting ways with the Cruz campaign on the news. Of Cruz, Vaden said, “I don’t entirely trust him.”

In Sparks, where Trump held a final pre-caucus rally Tuesday at the Nugget casino, some attendees visibly recoiled at the mention of Cruz’s name.

“It seems like every time that man opens his mouth, he lies,” said Linda Rengifo, 74, a retiree from Reno. “He fired that man for lying, but I’d be surprised if he didn’t put him up to it. If he was the nominee, I couldn’t vote for him.”

Sean Sullivan in Las Vegas and David Weigel in Sparks, Nev., contributed to this report.