Nabeel Gabol, a PPP leader, said: "If the major political party believes that he is the most talented person, then he is the most eligible person for this post.
"Now it depends on him, whether he himself becomes [president] or nominates someone else".
The nomination could, however, further destabilise Pakistan's fragile ruling coalition.
The government is struggling to tackle economic problems in the country, unrest in the tribal areas and the growing strength of armed pro-Taliban groups.
"We have asked [the PPP] to tell us by Saturday night whether or not judges can be restored on Monday or not"
Nawaz Sharif, leader of the PML-N
Nominating Zaradari also became mired in wrangling over the issue of the deposed judges.
Sharif, a two-time former prime minister whom polls suggest is Pakistan's most popular politician, has repeatedly threatened to go into opposition if the judges are not restored quickly.
Speaking on Saturday after a PPP delegation approached him to support Zardari, he said: "It had been agreed that when Musharraf would resign or get impeached, judges were to be reinstated automatically within 24 hours.
"We have asked them to tell us by Saturday night whether or not judges can be restored on Monday or not."
Pakistan's election commission announced that date of the presidential election on Friday, just four days after Musharraf resigned the office.
Kanwar Dilshad, the election commission secretary, said nomination papers for the presidency can be filed from August 26, with the final date for any withdrawals on August 30.
Under Pakistan's constitution, a president is elected by members of the country's four provincial assemblies and the national parliament within 30 days of the post becoming vacant.