The White House sought Tuesday to soothe concerns over a controversial birth control rule that has led to Republican attacks and tension among close allies of the administration.

The efforts followed mounting criticism from Catholics and other faith leaders that a new rule requiring certain religious institutions to cover contraception as part of their employee health plans violates their constitutional rights.

And they came as White House officials began hearing complaints from some of their own allies and advisers, who view the rule as a policy mistake that feeds what they see as an unfair charge from Republicans: that President Obama is anti-religion.

The administration’s response Tuesday came on two tracks — with officials telling liberal groups and lawmakers that they were not backing down, while trying to assure religious groups that a phase-in period will allow the two sides to agree on an approach to putting the rule into practice.

“There are conversations right now to arrange a meeting to talk with folks about how this policy can be nuanced,” said Joel C. Hunter, a Florida megachurch pastor who has grown personally close to Obama and advised his White House on religious issues. “This is so fixable, and we just want to get into the conversation.”

White House press secretary Jay Carney said Obama is taking the objections of the Catholic leaders “seriously” and will seek to implement the policy in a way that “allays some of those concerns.”

But the assurances were greeted Tuesday with skepticism from the U.S. Conference of Catholic Bishops, which has been leading the opposition to the new requirement.

“So far, ‘work this thing through’ is just the sugar-coated version of ‘force you to comply,’ ” Anthony R. Picarello Jr., general counsel for the conference, said in an e-mail.

Last week, priests around the country read from their pulpits strongly worded letters from Catholic leaders condemning the administration’s actions. And group officials said they have been told in no uncertain terms that there will be no reversing of the policy.

The events Tuesday underscored how the issue presents political challenges for Obama, who is trying to balance the demands of his liberal base against the need to court Catholics and other religious voters, particularly in swing states such as Ohio and Pennsylvania.

Supporters of the rule say that all employers, regardless of their religious affiliations, should be required to provide female employees the full breadth of health-care coverage, including birth control, the “morning-after pill” and sterilization services. The rule exempts churches but covers religiously affiliated colleges and hospitals, many of which receive federal funds.

But contraception services violate Catholic tenets, and the bishops, as well as leaders from other faith groups and denominations, say the rule violates their religious freedom.

Republicans see vulnerabilities for the president, not just on the issue of religious liberty but also on the role of government. They believe that Obama, in approving the new regulation, played into their long-running critiques that he is anti-religion and eager to expand the reach of government.

The issue provides a particular opportunity for the Republican presidential front-runner, Mitt Romney, who has struggled to connect with religious conservatives who are skeptical of his past positions and his Mormon faith. Romney’s campaign said Tuesday that he would repeal the new rule soon after taking office, and Romney charged in a speech Monday that the provision was a “violation of conscience.”

The response Tuesday from Obama aides suggested that they see at least the potential for a problem.

David Axelrod, a senior adviser to Obama’s reelection campaign, seemed to suggest early in the day on MSNBC’s “Morning Joe” that the White House might be open to a compromise on the matter.

“We certainly don’t want to abridge anyone’s religious freedoms, so we’re going to look for a way to move forward that both provides women with the preventive care that they need and respects the prerogatives of religious institutions,” he said.

His comments sparked immediate concern among Democratic lawmakers, who worried that the White House was poised to bend to its critics.

Sen. Barbara Boxer (Calif.), one of a number of Democratic lawmakers who had lobbied Obama to keep the religious exemption narrow, called Axelrod for assurances that his remarks reflected no change of heart. The presidential adviser was “very clear” that “this decision is the same as it was before,” Boxer said.

“We’re not going to back off the principle that women who work in these institutions should have the same access to contraception as others do,” Axelrod said in an e-mail to The Washington Post.

The brouhaha threatens to undermine extensive efforts on the part of Obama, starting during his 2008 campaign, to court religious voters — many of whom are political independents and were initially skeptical of him.

The result is a mixed record that has angered activists on the left and the right.

Liberal groups have been frustrated by Obama’s continuation of many Bush-era initiatives designed to funnel more money to faith-based groups. Notably, Obama has not rolled back a rule allowing groups that get federal grants to discriminate in their hiring practices.

“They have done things that are counter to the message that is being talked about in the media, that there’s a ‘war on religion,’ ” said Dena Sher, legislative counsel for the ACLU.

Staff writers David Nakamura, N.C. Aizenman and Michelle Boorstein and polling analyst Scott Clement contributed to this report.