Some congressional Democrats voiced a fresh round of concern on Sunday about Graham Platner, the party’s presumptive Senate nominee in Maine, while others reiterated their support for his candidacy after several women described volatile and “toxic” relationships with him.

The reactions this weekend — on Sunday morning news shows, social media and elsewhere — reflected a party grappling with Mr. Platner’s campaign in a key Senate battleground just before Tuesday’s primary election. Even one of Mr. Platner’s most vocal supporters, Representative Ro Khanna of California, said on Sunday that he believed the account of a woman who accused Mr. Platner of physically threatening behavior, and urged his campaign not to criticize her.

The New York Times reported on Thursday that three women who dated Mr. Platner had said that he could be volatile in relationships and demeaning to women. The report followed revelations on May 30 that Mr. Platner had exchanged sexual messages with women outside his marriage as recently as last year.

Senator Mark Warner of Virginia, a centrist Democrat, called the allegations “beyond disturbing” in an appearance Sunday on Fox News. But he said Maine voters would ultimately decide if they were disqualifying.

“If these allegations are true, they’re very troubling,” he said on “Fox News Sunday.” “But I don’t know if they are true or not. I think, frankly, the people in Maine ought to decide that.”

Mr. Platner “strongly disputes” any claims of physical intimidation or altercations, his campaign has said. A Marine Corps veteran who deployed to Iraq, Mr. Platner has described struggles with PTSD and said that he has “self-medicated with alcohol and was a far from perfect boyfriend” during what he described as a “very dark period” in his life that preceded a “transformation.” He dismissed reports about him sending sexual messages as distractions from issues voters care about.

Mr. Warner, who appeared on several Sunday shows, said that Mr. Platner’s behavior would probably have taken him out of the running in earlier decades, but that President Trump had lowered the bar for candidates’ personal conduct.

“President Trump set a new standard,” Mr. Warner told “This Week” on ABC News, invoking the “Access Hollywood” tapes in which Mr. Trump boasted about groping women. “Whether that low standard is what we ought to proceed with is going to be, again, in the hands of the voters.”

Representative Tom Suozzi of New York, a moderate Democrat, questioned Mr. Platner’s progressive policy stances and suggested that the reports about Mr. Platner’s personal life could put the party at risk of losing an important race in November.

“I’d say the allegations are unsettling,” Mr. Suozzi said in a phone interview Sunday morning. “I’d say some of his policy positions are unsettling. I’d say some of his past behavior is unsettling. And the effect that this will have on the final result in November is unsettling. It’s all very, very unsettling.”

Some prominent progressive lawmakers have stuck by Mr. Platner in the aftermath of the accounts of his behavior by the three women, a conservative and two Democrats. Mr. Khanna, who campaigned for him in Maine on Friday, said in an interview broadcast Sunday on CBS’s “Face The Nation” that while Mr. Platner’s actions were “misogynistic” and “shameful,” he believed he had taken accountability.

But Mr. Khanna also said he believed the account of Lyndsey Fifield, who dated Mr. Platner from roughly 2013 to 2015 and spoke to The Times about her experiences with him. Ms. Fifield said that Mr. Platner regularly grabbed her by the shoulders and once twisted her arm behind her back before shoving her into a bedroom during their relationship. Mr. Platner’s campaign called Ms. Fifield a “lifelong G.O.P. operative” in a statement responding to her allegations. Ms. Fifield is a Virginia conservative who has worked for right-leaning groups and Republican campaigns.

Mr. Khanna said that the Platner campaign “should not attack” Ms. Fifield and that her politics were “irrelevant.”

“Obviously, if there was any evidence that comes out that there is actual domestic violence or assault — I have zero tolerance for that,” Mr. Khanna said. He added, “Here you have someone who was in toxic relationships and was ashamed about it, who served his country, and the Maine voters are saying, ‘Look, let’s give him some grace.’”

The Platner campaign did not have any immediate comment on Sunday. Mr. Platner was scheduled to appear at a campaign event later in the day.

Maine is seen as a central battlefront in the fight for control of the Senate in the midterms. Mr. Platner became the presumptive Democratic nominee after Gov. Janet Mills, a Democrat, suspended her campaign for Senate in late April. He is expected to face Senator Susan M. Collins, who is running unopposed in the Republican primary.

Mr. Platner has generated plenty of excitement for his campaign, but has also faced scrutiny over his past, including over inflammatory online posts and a tattoo resembling a Nazi symbol. (He has said that he did not know of the symbol’s Nazi associations until last year, and that he removed the tattoo.)

Some allies of Mr. Platner have sought to redirect attention away from the latest allegations. Senator Bernie Sanders of Vermont, one of Mr. Platner’s best-known supporters, wrote Saturday on social media that Mr. Platner was the “only one” in the race who would act to address a “rigged economy,” “endless wars” and “a corrupt campaign finance system.”

“We cannot continue to allow the Republican Party to control the Senate and push forward Trump’s oligarchic and authoritarian agenda,” he wrote. “And there is only one candidate who will do something about it.”