Ashraf, 61, appeared before the court and requested that he be given four to six weeks to write the letter to Swiss authorities, reported Dawn.
"I need time to understand the matter… as it involves certain legal and constitutional points," Ashraf told the court. He also pleaded that the show-cause notice be withdrawn.
"I need enough time from the court to seek the assistance of constitutional and legal experts," he said.
Ashraf's predecessor, Yousuf Raza Gilani, was disqualified by the court for contempt after he refused to accept the court's orders.
During the Monday hearing, the apex court initially gave time till Sep 12 while Ashraf sought time till Sep 16 as he had professional commitments. The prime minister said he will be in China till Sep 14, therefore, more time should be granted, reported Geo News.
The Supreme Court ordered Ashraf to appear in the court Sep 18 and adjourned the hearing.
Jostled by crowds, Ashraf waved before stepping inside the court building, where there was tight security.
A five-judge special bench comprising Justices Asif Saeed Khan Khosa, Ejaz Afzal Khan, Ijaz Ahmed Chaudhry, Gulzar Ahmed and Muhammad Ather Saeed heard the case.
The apex court had issued the notice Aug 8.
The order said it was unfortunate that the prime minister had failed to implement the court's directives.
The court had July 25 given Ashraf time till Aug 8 to write to Swiss authorities to reopen corruption cases against Zardari.
Accused of graft, Zardari was granted amnesty under the National Reconciliation Ordinance (NRO) in 2007 by then president Pervez Musharraf to facilitate his return home from exile, and primarily that of his wife, former prime minister Benazir Bhutto.
Zardari and Bhutto were suspected of using Swiss accounts to launder about $12 million in alleged bribes paid by companies seeking customs inspection contracts in the 1990s.
The NRO that granted immunity to politicians and bureaucrats in corruption cases was struck down by the Supreme Court as void in 2009.
The apex court in January ordered Gilani to write a letter to the Swiss authorities to reopen cases against Zardari.
Gilani was convicted April 26 of contempt of court, and was disqualified as prime minister as well as parliament member June 19.