WASHINGTON — Hammered by Republicans and the Catholic Church, the White House hinted at compromise Tuesday as it struggled to calm an uproar caused by its rule requiring religious schools and hospitals to include birth control in their employees’ health insurance.

Obama’s chief spokesman and his top campaign strategist both said the administration was searching for ways to allay the concern of Roman Catholics who say the mandate would force them to violate their religious beliefs against contraception.

Senate Republican Leader Mitch McConnell said the policy was a “huge mistake” that the administration should reconsider. “And if they don’t, Congress will act,” McConnell said.

On the GOP presidential campaign trail, Mitt Romney accused Obama of an “assault on religion,” and Newt Gingrich called the rule an “attack on the Catholic Church.”

Obama’s spokesman defended the decision that prompted the flare-up, even as he raised the prospect of some adjustment.

He said that women working for church-affiliated employers must be able to get contraception, but he also made clear that the White House wants to accommodate the concerns of the employers who would be required to provide birth control coverage regardless of their religious beliefs.

“There are ways to, I think, help resolve this issue that ensures that we provide that important preventive service, that health care coverage, to all women … in a way that also tries to allay some of these concerns,” press secretary Jay Carney said.

The spokesman did not say what those ways might be.

Separately, Obama campaign strategist David Axelrod made the same point. “The real question is how do we get together and resolve this in a way that respects the concerns that have been raised but also assures women across this country that they’re going to have the preventive care that they need,” Axelrod said on MSNBC.

Axelrod’s and Carney’s comments created a sense that further accommodation would be considered by the White House, but there was no sign the administration would move to completely withdraw the rule.

Some Catholic supporters of the administration said they had noticed a shift in White House rhetoric that gave them hope for a compromise.

“It’s very different from what was said before, that this is final and nothing is going to change,” said Kristen Day, executive director of Democrats For Life of America.

Options for a compromise could include granting leeway for a church-affiliated employer not to cover birth control, provided it referred employees to an insurer who would provide the coverage.