"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 preventative care that they need and respects the prerogatives of religious institutions," David Axelrod, a top adviser to Obama's re-election campaign, told MSNBC.
"There are conversations right now to arrange a meeting to talk with folks about how this policy can be nuanced," the Rev. Joel C. Hunter, a Florida megachurch pastor and author of "A New Kind of Conservative" who has advised Obama on religious issues, told The Washington Post.
"This is so fixable, and we just want to get into the conversation."
But the U.S. Conference of Catholic Bishops -- among the most vocal critics of the rule, calling it an unprecedented abridgement of religious liberty -- said it had not yet been approached by the White House with any proposed changes.
While churches are exempt from the rule, issued Jan. 20 as part of the implementation of the 2010 healthcare overhaul, it requires some church-run hospitals, colleges and other institutions to provide free coverage for contraception to their employees. It also requires them to provide coverage for the so-called morning-after pill, taken after sexual intercourse, and sterilization measures such as vasectomies.
The administration offered groups a one-year grace period to figure out how to comply with the rules, but the bishops say they will pursue every legal measure to stop its implementation.
Pittsburgh Bishop David Zubik, who sits on the bishops conference National Advisory Council and its laity committee, told the Catholic News Agency the rule was a "slap in the face" to Catholics.
Axelrod told MSNBC similar rules exist in 28 states and Obama wanted to provide employees of the religiously affiliated institutions access to "the same package that every other woman in the country has, the same right and access to basic preventive care."
He called the rule a "violation of conscience," adding "we must have a president who is willing to protect America's first right -- our right to worship God according to the dictates of our own conscience."
A survey this week indicates 58 percent of U.S. Roman Catholics say employers should be required to provide employees with healthcare, including contraception.
The Public Religion Research Institute survey found women were significantly more likely to favor free contraception through employee healthcare plans, at 62 percent, versus 47 percent of men. Fifty-four percent of women agreed religiously affiliated colleges and hospitals should provide this coverage versus 43 percent of men.
The phone survey of 1,009 U.S. adults conducted Wednesday through Sunday has a margin of error of 3.5. percentage points.
Notable deaths of 2014 [PHOTOS]