ISLAMABAD, Pakistan (CNN) -- Leaders of Pakistan's ruling coalition were meeting Tuesday to discuss who will replace President Pervez Musharraf, who resigned Monday.
They also were to talk about how to restore dozens of judges he fired last year.
Under Pakistan's constitution, parliament has to elect a new president within 30 days. The nominee will need a majority vote of both houses of parliament as well as Pakistan's four provincial assemblies.
For now, Mohammedmian Soomro, chairman of the senate, is acting as Pakistan's caretaker president, but he is not expected to retain the position. Many within the ruling coalition see him as a Musharraf loyalist, analysts say.
Unless the two major parties in the ruling coalition can reach a consensus soon -- particularly on the issue of the judges -- a stalemate in the already fragile coalition could trigger another political crisis, analysts said.
The country's labor minister told reporters the coalition may announce a presidential pick by the end of the week. But, he said, a decision on the judges is expected earlier.
Pakistan's ambassador to the United States, Husain Haqqani, said Tuesday that Pakistan's presidency may eventually alternate between the ruling coalition's top two leaders: former Prime Minister Nawaz Sharif and Asif Ali Zardari, the widower of former Prime Minister Benazir Bhutto.
"One of them, and I will not say which one, said to me not long ago, 'You know what, in other democracies people take turns, why can't we?' " Haqqani told reporters.
He noted both men are young enough that they can be patient as his country implements democratic reform.
Zardari, the head of the one of the coalition parties, the Pakistan's People Party, is 52. Sharif, the leader of the other party in the coalition, the Pakistan Muslim League-N, is 58.
Haqqani said the presidential powers may be scaled back to become more ceremonial. He also said Musharraf's resignation may result in an improvement in relations between the United States and Pakistan.
"Everybody who disliked President Musharraf ended up disliking the United States," Haqqani said. "I think that has changed yesterday."
Musharraf -- who seized power from Sharif in a bloodless coup in 1999 -- quit just as the ruling coalition was taking steps to remove him in what would have been the first impeachment of a president in Pakistan's history.
The country's stock market and its currency rose after Musharraf stepped down on hopes that the nuclear-armed Islamic nation of 164 million was headed for political stability.
PPP, the ruling party, has not ruled out the possibility of granting him "safe passage" out of the country. That would be a kind of payback for Musharraf's decision last year to sign an amnesty that cleared Bhutto, then the PPP leader, from prosecution after she returned home to campaign for prime minister, a post she already held twice.
Bhutto had been in an eight-year self-imposed exile after being convicted on money laundering charges. She was assassinated late last year in Pakistan while campaigning.
The other party in the coalition, the PML-N, has publicly stated its opposition to amnesty for Musharraf. Its leader, Sharif, wants the former president tried on charges that the coalition compiled during its attempt to oust him.
Haqqani declined to predict whether Musharraf will be granted safe passage outside of Pakistan, saying that will be a matter for Parliament to decide. He did say, however, that he felt the days of putting ousted political leaders in prison are over.
The coalition partners also differ on the issue of the judges.
Last November, Musharraf declared a state of emergency and fired some 60 judges, including 14 of 18 who sat on the Supreme Court.
Some say Musharraf sacked the judges because they were set to rule against the legitimacy of his third term in office. He had been re-elected president by a parliament stacked with his supporters, they said.
After sweeping into power in parliamentary elections in February, the coalition promised to reinstate the judges within 30 days of taking office, but that hasn't happened.
Sharif's central election promise was to depose Musharraf and put the judges back on the bench. It was also his condition for joining the coalition. He believes a simple resolution followed by an executive order will be enough to restore the judges. But the PPP wants constitutional changes along with a resolution. That would take longer.
One reason behind the delay, some experts say, may be that the Supreme Court was expected to look into the controversial amnesty granted to Bhutto and her husband, Zardari, for the corruption charges.
CNN's Alizah Kassim in Atlanta, Georgia, and Reza Sayah in Islamabad, Pakistan, contributed to this report.