"[Do] what [President] Obama never did - Obama never had proper interaction with any Pakistani chief executive," he said.
Zardari, who met with Obama at the White House in 2011, said he was not interested in openly criticising Obama.
However, his comments sounded at times like an indictment of Obama's handling of what has, for more than two decades, been one of Washington's most vexing and complex foreign relationships, Washington Times said.
Zardari said what is needed today is a US president who can breathe new life into the relationship by realising that the wars against extremism in Afghanistan and Pakistan are the same - and are no different from those being waged against the Islamic State and al Qaeda-aligned groups from Syria and Iraq to Yemen and Libya, the paper said.
More From This Section
"We're losing the battle of minds against extremists in Afghanistan, we've lost it in Pakistan, we've lost it in Syria, we've lost it in Yemen, we've lost it in Iraq, we've lost it Libya, we've lost it everywhere," the 61-year-old former president was quoted as saying by the paper.
"That would entail, basically, confidence-building between the different countries in the region and meaningful actions" toward defeating the extremist mindset.
Trump should appoint an envoy like Richard C Holbrooke early in his term and not allow Washington partisanship to influence the move, Zardari said.
The former Pakistani president suggested that a good pick might be Anne W Patterson, a career Foreign Service officer and a former ambassador to Pakistan and Egypt.
Zardari, the only elected Pakistani president ever to have completed a full term in office, said that Trump's critics should not dismiss him out of hand but rather give him a chance to make his mark in the region, beset by conflicts in Afghanistan, jihadi terrorist movements and the increasingly tense India-Pakistan conflict over Kashmir.
Zardari, co-chairman of the left-leaning Pakistan People's Party, was in Washington to participate in Trump's inaugural festivities.