OSHKOSH, Wis. — Donald Trump’s latest rude comments about Ted Cruz’s wife are raising new alarms among Republicans about the party front-runner’s ability to win over women, especially in a potential fall presidential match-up with Hillary Clinton.

Trump is under fire for jabs at Heidi Cruz, as the rivals engage in an increasingly bitter, personal battle for the GOP presidential nomination. Hostilities reached a new high Friday when Cruz accused Trump and “his henchmen” of stoking false rumors that he’d cheated on his wife.

“We don’t want a president who traffics in sleaze and slime,” the Texas senator told reporters in Wisconsin. “We don’t want a president who seems to have a real issue with strong women.”

Trump’s history of sexist comments, from his “Apprentice” television program to racy interviews with radio host Howard Stern, have long been seen by Republicans as a potential vulnerability, especially in a general election match-up with Clinton, who would be the country’s first female president.

The issue took off in the first GOP debate when Fox News’ Megyn Kelly asked Trump about calling women “fat pig,” “dog” and other names. Her question sparked a continuing quarrel between Trump and the network.

Trump also faced a backlash after he was quoted in a “Rolling Stone” profile insulting businesswoman Carly Fiorina, who endorsed Cruz after she dropped out of the 2016 Republican race.

“Look at that face!” Trump was quoted saying. “Would anyone vote for that? Can you imagine that, the face of our next president?!”

The issue reignited Wednesday after an anti-Trump super PAC released an ad featuring a risque photo of his wife, Melania, a former model, taken in a GQ photo shoot.

“Meet Melania Trump. Your Next First Lady. Or, you could support Ted Cruz on Tuesday,” it read. Trump responded by falsely accusing Cruz of running the ad and warning, “Be careful, Lyin’ Ted, or I will spill the beans on your wife!” Cruz’s wife is a former Goldman Sachs investment manager and White House aide, who served as economic policy adviser to President George W. Bush.

On Wednesday, Trump escalated things when he retweeted side-by-side images of Cruz’s wife, with an unflattering grimace, and Melania in a gauzy, glamorous pose. “No need to ‘spill the beans”’ read the caption. “The images are worth a thousand words.”

At an event Thursday in Wisconsin, Cruz called Trump “a sniveling coward” who has a problem with “strong women.”