But they failed to find common ground on issues including relations with the EU, institutional reform and fishing.
The president then called on the Left-Green Movement, the second-biggest party, to form a government.
Despite holding talks to build a five-party coalition from the centre-right to the far-left, disagreements over taxes and other issues led the negotiations to collapse in late November.
The president then allowed the parties to hold informal talks, which led the Independence party and the Left-Green Movement to discuss terms for sharing power. But the diametrically opposed parties could not find enough common ground.
Giving the Pirate party, which came third in the election, the chance to build a government has been seen as a bold move that is not guaranteed to be a success.
“I am optimistic that we will find a way to work together,” Jónsdóttir said.
The scandal over the Panama Papers, released in April, ensnared several Icelandic officials and led to the resignation of former prime minister Sigmundur Daviíð Gunnlaugsson, prompting the October vote.
With voters keen to see political change, the small and controversial Pirate party had vowed during the election campaign to implement radical institutional reforms for more direct democracy and greater transparency in public life.
It won 14.5% of votes, less than pollsters had predicted.