Senior aides of the Pakistani president, Asif Ali Zardari, have said that he will remain out of Pakistan for up to two weeks following a "small stroke" eight days ago, raising fears of a long period of political instability in the south Asian state.

An associate of the president told Associated Press that the 56-year-old politician would stay in the Gulf emirate of Dubai for further medical tests but insisted there was no question of Zardari being too ill to return to office.

Fauzia Wahab, an adviser to the president and former information secretary, told the Guardian that Zardari "remained and would remain in power" and would be back "soon". However, the president and his close circle remain worried about the possibility of a military coup.

Much attention is now being focused on Zardari's son, Bilawal Bhutto-Zardari, who spent much of his youth overseas but returned to Pakistan last year. "Bilawal is here [in Pakistan] to show that the family and the party are going nowhere. There is no question of anyone running away," Wahab said.

A plan for the 23-year-old to participate in a series of rallies and visits across Pakistan in coming weeks has been put on hold due to security concerns, but it is clear that the recent Oxford graduate, whose mother, the former prime minister Benazir Bhutto, was assassinated four years ago, is being prepared for power.

"His father has only ever seen himself as a transition leader. Bilawal is the blood heir to a family of political martyrs so he is the political future," said one observer close to the Bhutto family.

A "mass contact campaign" to launch Bilawal, who succeeded his mother as chairman of the Pakistan People's party (PPP), as a major national figure was mooted by senior figures in the organisation last month.

Bilawal will play a major role in a big commemoration of the death of Benazir, killed on 27 December 2007. Since Zardari's departure for Dubai, every engagement of his son has been publicised by government and party spokesmen.

Bilawal's agenda resembles that of a head of state, rather than a recent history graduate. Over two days last week, he had meetings with the top leadership and young activists of the ruling PPP, was briefed by a top parliamentarian on national security and held series of receptions held for the PPP's coalition allies.

He also issued an order that the party follow a harsher line on US-Pakistan relations, currently at a nadir following the death of 24 Pakistani soldiers in a border clash with Nato troops, officials said.

Recent days have been filled with similar public and political appointments. "It is inevitable that when they are feeling insecure – as many senior PPP people and particularly the close circle around the president do right now – they rally around their greatest asset," said Prof Osama Siddique, a political scientist in Lahore.

Since Zardari took power after the resignation of the former military general Pervez Musharraf in 2008, deep economic problems, extremist violence and events such as the US special forces raid that killed Osama bin Laden in May have put his administration under immense pressure.

Even before the recent publication of a memo supposedly penned by a key ally, the ambassador in Washington, asking for US help to hold off a military coup following the Bin Laden raid, relations with the powerful Pakistani security establishment were poor.

Senior judges are now probing the memo and have ordered that corruption investigations overseas be reopened. Some suggest Zardari is staying out of Pakistan to avoid a confrontation after being summoned to give evidence before the supreme court in the memo inquiry.

However, the rumours gripping Pakistan of an imminent army coup were "overblown", said Siddique. "Zardari remains the elected president and in this day and age, any attempt to unseat him by coercive means is likely to backfire," he said.

Once dismissed as a political lightweight and dogged by graft charges, Zardari has consistently outmanoeuvred political opponents. The PPP, which he leads in tandem with his son, heads a broad coalition.

Though he is still too young to stand for parliament, the eventual succession of Bilawal appears inevitable. His grandfather, Zulfiqar Ali Bhutto, founded the PPP, became one of the country's most controversial leaders and, after being ousted, was hanged on the orders of the military dictator General Zia-ul-Haq, who usurped him from power.

Bilawal was educated at private schools in Pakistan and, after his mother went into self-imposed exile in 1999, in Dubai. Appointed party chairman in accordance with his mother's wishes, he remained in the UK until he graduated last year. Until now he had has kept a relatively low profile.

"The point is not that he is 23 years old or head of a political party but is the blood price that the Bhuttos have paid for power," said one observer. "Bilawal has accepted the possibility, even certainty, of a violent premature death."

Wahab, a presidential aide and Bhutto family friend, said Bilawal carried "a heavy burden". "He has the Bhutto genes," she said.