With the White House under fire for its new rule requiring employers including religious organizations to offer health insurance that fully covers birth control coverage, ABC News has learned that later today the White House — possibly President Obama himself — will likely announce an attempt to accommodate these religious groups.

The move, based on state models, will almost certainly not satisfy bishops and other religious leaders since it will preserve the goal of women employees having their birth control fully covered by health insurance.

Sources say it will be respectful of religious beliefs but will not back off from that goal, which many religious leaders oppose since birth control is in violation of their religious beliefs.

One source familiar with the decision described the accommodation as “Hawaii-plus,” insisting that it’s better than the Hawaii plan — for both sides.

In Hawaii the employer is responsible for referring employees to places where they can obtain the contraception; Catholic leaders call that material cooperation with evil. But what the White House will likely announce later today is that the relationship between the religious employer and the insurance company will not need to have any component involving contraception.