The French president asked Elisabeth Borne to stay on as prime minister, and has been meeting political opponents to seek parliamentary support after his coalition lost its majority.
France's President Emmanuel Macron has held talks with his rivals on Tuesday, to try and find "constructive solutions" to form a working government, according to the Elysée.
Some of his allies have advocated a "grand coalition" or a government of "national unity" involving other parties.
Macron will likely need the support of one or more smaller parties to get his key reforms through the National Assembly, after losing his overall majority in Sunday's second round of legislative voting.
Six of Macron's political opponents arrived at the Elysée Palace one after the other on Tuesday, including the far-right leader Marine Le Pen.
More talks are due to take place on Wednesday but so far Jean-Luc Mélenchon — the leader of the next biggest bloc in the National Assembly, of far-left, socialist and green patries — is not yet expected to make an appearance. However, other leading figures from the movement are due to meet the president.
Prime Minister to remain in place
Prime Minister Elisabeth Borne offered her resignation on Tuesday — as a number of party leaders had demanded — but this was refused by President Macron whose office said there was a need for her to assure the government's day-to-day operation for now.
The decision was strongly criticised by some opposing parties, particularly among Mélenchon's "La France Insoumise" ("France Unbowed") party, as well as on the right.
"It is quite spectacular to see that Emmanuel Macron does not seem to draw any consequences from the election" and "his crushing electoral defeat", LFI's number two Adrien Quatennens told the National Assembly.
Jean-Luc Mélenchon called on Tuesday for Borne to submit to a vote of confidence from members of parliament, saying that the head of government had "no legitimacy" to govern.
Elisabeth Borne, a former minister and government official who heralds from the centre-left, was appointed by Macron in the aftermath of his presidential victory in April. The move was seen as a gesture to left-wing voters who had backed him in the second round against Le Pen.
Calls for 'national unity' government
Among Emmanuel Macron's allies, the centrist MoDem leader François Bayrou called for a formation "as close as possible to national unity".
Former Prime Minister Edouard Philippe urged the creation of a "grand coalition", considering a "compromise" with the Republicans and negotiating with the Socialist Party possible, even if it meant "finding new profiles" to take the lead.
According to Fabien Roussel of the French Communist Party (PCF), who met Macron on Tuesday evening, the president asked him "if we were ready to work in a government of national unity", and if such an initiative was the right solution or whether support should be sought "on a case-by-case basis".
Marine Le Pen of the National Rally said her party was claiming two influential posts, the presidency of the finance committee and the vice-presidency of the National Assembly. The far-right group intended to be "in opposition" but "not in systematic obstruction", she added.
'No question of coalition pact', say Republicans
Macron's lack of a majority could amplify the right-wing Republicans' influence as the president's group seeks votes in parliament. The party is the fourth largest in parliament with 61 seats.
Party leader Christian Jacob, the first to meet Emmanuel Macron on Tuesday, refused to enter into any pact with the government — a move he said would amount to "a betrayal of our voters".
"We remain in opposition in a determined but responsible way," he said afterwards. "We will never block the institutions but there is no question of entering into a coalition pact."
He added that it was up to the president "to put proposals on the table".
Jacob was succeeded by the Socialist Olivier Faure, who told Macron his party was "willing" to "move forward" if the government took measures on spending power, in particular a rise in the minimum wage.
'Imagination' needed to govern
At the end of the second round of elections on Sunday, Macron's candidates won 245 seats, ahead of the left-wing coalition Nouvelle Union Populaire Ecologique et Sociale (NUPES) with 131 and the far-right National Rally which made a historic breakthrough with 89 seats.
The president's "Ensemble" alliance fell well short of the 289 seats needed for an overall majority.
Macron’s camp might have to contend with ruling with a minority in parliament or cohabiting with a prime minister and government from a different camp.
They now face the realities of a National Assembly with two hostile opposition groups as he tries to push through important reforms - including on pensions, tax cuts and raising the retirement age - all of which will be nearly impossible to do without a coalition partner.
"We will have to show a lot of imagination" to govern, admitted economy minister Bruno Le Maire.
The left-green NUPES bloc will most likely strongly oppose Macron’s reforms, having platformed on actually lowering the retirement age from 62 to 60 and return of wealth taxes on people and companies. However bloc leader Jean-Luc Mélenchon will most likely have to give up on his ambition of being prime minister.
Mélenchon vowed to keep the bloc together, branding it a “rebellion [which] now has a face”. However, his call on Monday for his alliance to form one parliamentary group immediately ran into trouble, with the Socialists, Greens and Communists all rejecting the idea.
Following the results, National Rally leader Marine Le Pen vowed to "implement the blocking of all harmful reforms" that Macron is proposing, "first and foremost" plans to raise the retirement age in France to 65.