Zardari made the announcement while addressing PPP parliamentarians and workers at the presidency late on Wednesday night.
The meeting was also attended by several federal ministers, including Commerce Minister Makhdoom Amin Fahim and Interior Minister Rehman Malik.
Following his mother Benazir Bhutto's assassination in December 2007, Bilawal was named as chairman of the PPP. His father Zardari was appointed as co-chariman of the party.
Bilawal has said that he would enter politics after he completes his education in Britain.
Zardari's announcement appears to have been made with an eye on the general election scheduled for 2013.
The PPP-led government, after completing more than half of its tenure, has now "entered the election mode" and party workers and lawmakers should enhance contacts with voters in their constituencies, he said.
Referring to the PPP-led government's performance, Zardari told the meeting: "We have had no smooth sailing and despite negative propaganda against the government and the party, we have not only survived but waded through gigantic challenges successfully and achieved significant achievements on all fronts."
The PPP will "continue forward with the march and conquer all the challenges", he said.
PPP leaders and workers have to reach out to the people in the two remaining years of the government's tenure and "also make new political alignments", he added.
Zardari urged the parliamentarians and workers to dedicate themselves to the welfare of the people as the government's performance will determine the PPP's position in the next general election.
The meeting discussed the results of by-elections in Pakistan's Punjab province and the lessons learnt from them.
The lawmakers and workers spoke on a host of issues, including the current political situation, coalition matters, alliance adjustment and party organisation.
The meeting reposed "full confidence" in the leadership of Zardari, presidential spokesman Farhatullah Babar said.
Image: Bilawal Bhutto