Bush Administration officials tell CBS News White House correspondent Jim Axelrod that they are growing more confident that Abdul Rahman will not be executed.
Pressure against the case has been building, and the Afghan government may be rethinking the charges against Abdul Rahman.
"He could be released soon," an Afghan government official told The Associated Press on condition of anonymity because he was not authorized to comment on the case to the media.
Look for Rahman to be freed before Tuesday, Axelrod reports, because Congress is now considering a request from the Bush administration for an additional $2.5 billion this year in aid to Afghanistan.
Tuesday is when U.S. Secretary of State Condoleezza Rice is expected to go to Capitol Hill to make the case for that money.
Senior clerics in the Afghan capital have voiced strong support for prosecuting Rahman and again warned Friday they would incite people to kill him unless he reverted to Islam.
Ansarullah Mawlavi Zada, the chief among three judges trying the case, asserted the autonomy of the court.
"We have constitution and law here. Nobody has the right to put pressure on us," he told the AP.
Australia's Prime Minister John Howard, meanwhile, joined the chorus of Western leaders expressing outrage over the prosecution and said he would protest personally to President Hamid Karzai.
"This is appalling. When I saw the report about this I felt sick literally," Howard told an Australian radio network Friday. "The idea that a person could be punished because of their religious belief and the idea they might be executed is just beyond belief."
Rahman faces the death penalty under Afghanistan's Islamic laws for converting 16 years ago while working as a medical aid worker for an international Christian group helping Afghan refugees in Pakistan.
Karzai's office has declined to comment on the case. Karzai is in a tough spot as pressure mounts from both the West to free Rahman and internally from Muslic clerics, who say they want Rahman dead.
Karzai took power after the ouster of the hard-line Taliban regime in a U.S.-led war in late 2001 and relies on international forces to maintain his still-shaky grip on the country.
But he would be reluctant to offend Islamic sensibilities at home or alienate religious conservatives wielding considerable power.
Diplomats have said the Afghan government is searching for a way to drop the case. On Wednesday, authorities said Rahman is suspected of being mentally ill and would undergo psychological examinations to see whether he is fit to stand trial.
Rice phoned Karzai on Thursday, seeking a "favorable resolution" of the case. She said Washington looked forward to that "in the very near future."
Senior clerics condemned Rahman as an apostate.
Rahman had "committed the greatest sin" by converting to Christianity and deserved to be killed, cleric Abdul Raoulf said in a sermon Friday at Herati Mosque.
"God's way is the right way, and this man whose name is Abdul Rahman is an apostate," he told about 150 worshippers.
Another cleric, Ayatullah Asife Muhseni, told a gathering of preachers and intellectuals at a Kabul hotel that the Afghan president had no right to overturn the punishment of an apostate.
He also demanded that clerics be able to question Rahman in jail to discover why he had converted to Christianity. He suggested it could have been the result of a conspiracy by Western nations or Jews.
At a fruit market in Kabul, many ordinary Afghans said they supported the death penalty, but some wanted more investigation before meting out the punishment.
"In the past 30 years, so many Afghans have been killed in name of communism, Taliban and politics or for robbery. It's enough Afghans killed," said Ghulam Mohammed, 45, a former army officer. Clerics should talk to him (Rahman) and bring him to the right way."