Basescu could nominate someone else, but his choice would have to be approved by Parliament. If his candidate was rejected twice, Parliament could be dissolved and new elections called.
The government has threatened to try to impeach Basescu again if he refuses to nominate Ponta.
Basescu has not spoken since the election. On Monday, he was in Norway along with other European Union leaders to collect the bloc's 2012 Nobel Peace Prize for fostering peace on a continent ravaged by war.
Election official Marian Muhulet said Basescu's group won less than 17 percent of the vote. A populist party led by a media tycoon scored about 14 percent and an ethnic Hungarian party won just over 5 percent. Other parties did not get the minimum 5 percent.
Basescu's allies in government grew unpopular due to the harsh austerity measures — spending cuts and tax hikes — they imposed and allegations of cronyism. Ponta was appointed prime minister in May after a confidence vote — making him the third prime minister this year.
Ponta restored most pensions and salaries that were slashed as part of the bailout loan agreement, but has largely continued the policies of the previous Basescu-allied governments, including a 24 percent sales tax, one of the highest in the 27-nation EU.
In Romania, the job of prime minister entails running the country and distributing public finances, while the president names the chiefs of the intelligence services, appoints ambassadors and chairs the country's top defense body, the Supreme Defense Council.
Since his election as president in 2004 and re-election in 2009, Basescu has represented Romania at the EU and other international summits, which has caused friction with Ponta.