By James Oliphant and Amy Tennery

CLEVELAND (Reuters) - Donald Trump's tough talk on law and order in a time of growing national insecurity appears to be winning over some of his fiercest skeptics - women initially put off by his swaggering tone, his clashes with female critics and past affairs.

But so far those are mostly conservative women. As Trump hones his message, the challenge is to make it also resonate among independent or undecided female voters who are crucial to his hopes of winning the U.S. presidential election in November.

Turmoil on the streets both abroad and at home could give Trump a new opportunity to do just that.

On Monday, Trump tried to seize it with an evening program at the Republican National Convention with several women speakers, including his wife Melania, focused on security. She told the convention that her husband would offer the country new leadership and keep it "safe and secure."

Americans have been rattled by recent attacks in France and Florida, the murder of police officers in Texas and Louisiana, and widespread protests over the killings of unarmed black men, polls show.

“We’re not electing a husband, we’re not electing a preacher, we’re electing a leader,” said Kay White, a Republican delegate to the nominating convention from Tennessee who originally supported U.S. Senator Ted Cruz of Texas in the early nominating contests, or primaries.

For White, it’s a one-issue election. “Security,” she said. “Nothing else matters.”

A year ago, Penny Nance, president of Concerned Women for America, a conservative advocacy group, was a blistering critic of the thrice-married Trump because of comments he made about Fox News anchor Megyn Kelly that many felt were sexist.

But Nance now supports Trump and says the candidate can fashion an effective message for women centered on national security, she told Reuters on the sidelines of the convention. “Every day we wake up and get our kids ready for school and the television is on and there is another attack,” she said.

"SECURITY MOMS"

For Trump, independents will be a tougher sell. A majority of women who identify as “independent” continue to hold an unfavorable opinion of him.

According to a Reuters/Ipsos opinion poll from June 1 to July 15, 64 percent of independent women voters expressed a “very unfavorable” or “somewhat unfavorable” view of Trump. That compares with 89 percent of Democratic women voters and 31 percent of Republican women voters who have a similarly unfavorable view of the New York businessman.

Trump isn't the first candidate whose tough talk on security has won over women voters. George W. Bush used the same message in his re-election campaign in 2004, in the midst of his "war on terror," to appeal to so-called security moms who were concerned about terrorism.

Bush garnered 48 percent of the vote of American women in that election. By comparison, Republican candidate Mitt Romney drew 44 percent in 2012, a full 10 points behind President Barack Obama.

Trump has had trouble convincing women nationally to support him ever since he entered the race in the summer of 2015, according to Reuters/Ipsos polling. Among women who are expected to vote on Nov. 8, a majority expressed an unfavorable opinion of him every time they were asked in the polls.

This month, women appeared to have soured even more on the likely Republican nominee: 69 percent of likely women voters expressed an unfavorable opinion of Trump in the latest July 11-15 Reuters/Ipsos polling, up from 63 percent in the five-day poll that ended on July 1.

Jen Lawless, director of the Women and Politics Institute at American University, said she doubted Trump could rehabilitate his image with most women voters, saying his conflict with Kelly, in particular, had lasting damage.

“I think that ship has sailed,” Lawless said.

Trump’s problems with women were dramatized on Monday when Women for Trump, a voter-support group, held a sparsely attended event in Cleveland.

The moderator, Jennifer Hulsey, appeared to speak for a lot of women who have found a way to support Trump. “He’s not perfect, but neither am I,” Hulsey said.

Some of the women delegates interviewed by Reuters at the convention on Monday were scornful of the criticism of Trump's attitude toward women, pointing to his daughter Ivanka, who is a top executive in the Trump Organization.

Ivanka and his other daughter Tiffany are due to speak at the convention this week.

"He was not my first choice or even my second choice," said Carol Del Carlo, a Trump supporter. But, "I look at him and I see how he treats his daughters. They’re engaged, they look like a real, true family, they work in their father’s business, they’re advisors."

(Reporting by Michelle Conlin, Emily Flitter, Ginger Gibson, Luciana Lopez, Emily Stephenson, and Amy Tennery; Writing by James Oliphant, editing by Paul Thomasch and Ross Colvin)