WASHINGTON » With the cameras running and the microphones on, congressional Democrats express outrage over Republican efforts to limit the types of health care that employers have to offer to their workers, particularly contraception. This is a fight Democrats are perfectly pleased to have.

As the issue of contraception access comes to the Senate this week, White House officials and Senate Democrats are increasingly hopeful that it will cut in their favor, believing that voters will conclude that Republicans are overreaching under the rubric of religious freedom. Democratic leaders, who set the Senate floor schedule, plan to hold a vote this week on a measure offered by Sen. Roy Blunt, R-Mo., that would in effect reduce insurance coverage of contraception, by allowing religious institutions not to cover it in the health plans they offer employees.

Democrats see the vote as a way to embarrass Republicans — especially those up for re-election in more moderate states, like Maine and Massachusetts — and believe the battle may alienate women and moderates from the Republican Party. Republicans need to pick up a number of seats to take back the Senate.

"They’ve gone way overboard in the mind of independents," Sen. Charles E. Schumer of New York, the No. 3 Democrat in the Senate, said in a conference call with reporters, referring to Republicans generally. The fight over contraception, he said, "is going to do lasting damage," to the Republican Party.

But Republicans contend that the issue helps them politically, because it highlights what they see as President Barack Obama’s hostility to religious freedom. "Americans fundamentally understand that an attack on religious liberty by the federal government is an attack on our most basic, personal freedoms," said Kevin Smith, a spokesman for House Speaker John Boehner, "and Democrats risk alienating millions of Americans if they continue down this path."

The Democrats’ confidence is a turnabout from a few weeks ago, when they had become worried that Obama might be alienating religious voters. Under pressure from Catholic groups, he modified the policy, saying Catholic institutions would not have to pay for the birth control coverage or refer their employees to it. But they still must offer plans that cover contraception, with the insurance companies covering the direct costs.

Some polls show Americans roughly divided over the issue of faith-based employers and contraception, while others have demonstrated an edge for Obama. In the latest /CBS News poll this month, 61 percent of those polled said they supported the Obama administration’s policy. Blunt’s measure would allow employers or insurance plans to exclude any health service that runs counter to their religious or ethical beliefs. The measure is in response to the Obama administration’s change to a provision in the health care overhaul passed in 2010 that requires employers to offer preventive care, including free birth control.

But for many Republicans, the compromise did not go far enough, and several have signed on to Blunt’s measure, saying the Obama administration was not respecting religious freedom with its rule. House Republicans are weighing their own bill in response to the administration’s rule.

The issue of women’s health care in particular is resonating nationwide. State legislatures, largely those under Republican control, are revisiting their laws concerning insurance coverage, contraception and abortion. In Virginia last week, Republican lawmakers passed a bill that would force a woman to receive an ultrasound before having an abortion, as well as require doctors to ask whether she wanted to hear the fetal heartbeat and obtain a printed image of the fetus. But the governor has wavered on the measure.

Senate Republicans who support Blunt’s bill say that they are not concerned with contraception per se, but with protecting religious freedom, a position they believe most Americans share. "This is not a women’s rights issue," Sen. Kelly Ayotte, R-N.H., said during a recent news conference with Blunt. "This is a religious liberty issue."

In an op-ed article in the St. Louis Post Dispatch over the weekend, Blunt framed his amendment as a correction to the 2009 health care law, which made broader requirements on coverage. "We have a responsibility to project those liberties from government intrusion," he wrote, "and I will continue to work with my colleagues on both sides of the aisle to ensure that the Obama administration’s unlawful health care mandate is repealed as soon as possible and replaced with the common-sense reforms our health care system needs."

At the White House on Monday, press secretary Jay Carney called Blunt’s legislation "dangerous and wrong," continuing a campaign to present the president’s stance as measured and reasonable. While many Democrats in the Senate and the White House, including Vice President Joe Biden, viewed Obama’s initial position on the contraception matter as a political mistake that unnecessarily alienated Catholic moderates, the administration’s later compromise has won wide support from the party and liberal Catholics.

Biden now believes the president "has landed in the right place," one senior administration official said.

Stephanie Cutter, Obama’s deputy campaign manager, said, "Red flags must be going up all over the Republican Party right now." She added: "If they’re not, they should be. Whether it’s the Virginia ultrasound bill, wanting to give bosses control over female employees’ health decisions, or the ongoing debates over personhood, which places an outright ban on birth control, they’re driving independent women away."

But lawmakers are on guard from either side. For instance, according to news reports, Rep. Kathy Hochul of New York was heckled at a town hall style meeting last week in her home state over the issue and at least one attendee carried a sign that read: "Kathy, why have you betrayed our Catholic institutions?"

One White House official cautioned that should the debate devolve into shrill arguments, the net result would be the alienation of independent or moderate voters whom Obama is trying to woo in his re-election bid. "Look, we don’t want to overplay this either, so we’ll be cautious," another White House official said.