President Emmanuel Macron on Monday asked his Prime Minister Gabriel Attal to continue handling day-to-day affairs, less than three weeks before the start of the Paris Olympics.
Macron leaves on Wednesday for a NATO summit in Washington.
Sunday's election left no faction even close to the majority needed to form a government, raising the risk of paralysis for the European Union's second-largest economy.
While a fractured parliament is not uncommon in Europe, the situation is unprecedented in France's modern history.
Leaders of the leftist coalition, the New Popular Front, which won the most seats, say they should form a new government.
The three main parties in the coalition - the hard-left France Unbowed, the Socialists and the Greens - began negotiations to find a candidate for prime minister.
Their talks are complicated by internal divisions.
Some are pushing for a hard-left figure while others, closer to the centre-left, prefer a more consensual personality.
France's prime minister is accountable to parliament and can be ousted through a no-confidence vote.
Members of France Unbowed and the Greens arrived on Tuesday morning at the National Assembly.
The Socialist MPs, who include former President Francois Hollande, were to gather in the afternoon.
The top negotiator for the Socialist party, Johanna Rolland, said that the future prime minister would not be Jean-Luc Mélenchon, the divisive hard-left founder of France Unbowed who has angered many moderates.
Speaking on France 2 television on Tuesday, she suggested the leftist coalition could possibly work with centre-left members of Macron's alliance.
"We will be open," she said.
According to official results, all three main blocs fell far short of the 289 seats needed to control the 577-seat National Assembly, the most powerful of France's two legislative chambers.
The results showed just over 180 seats for the New Popular Front leftist coalition, which placed first to beat Macron's centrist alliance, with more than 160 seats.
The far-right National Rally part of Marine Le Pen and its allies were restricted to third place, although their more than 140 seats were still way ahead of the party's previous best showing of 89 seats in 2022.
Macron has three years remaining on his presidential term.
