Italian media said claims that Conte had exaggerated his CV had delayed President Sergio Mattarella's decision on whether to approve him as premier.
Media also reported there were worries over the choice of an anti-euro economy minister by the parties, whose combative stance towards Brussels has caused fears for European financial stability.
But Mattarella's office later said Conte was summoned to the presidential palace at 1530 GMT.
The anti-establishment Five Star Movement (M5S) and the anti-immigrant League nominated Conte, 54, for premier on Monday. But scandal soon struck over doubts about his claims to have studied at certain top world universities.
"Conte betrayed by his CV," ran a headline in left-leaning newspaper La Republicca.
"The CV affair is open, Conte is hanging in the balance," said Il Corriere della Sera.
The parties are seeking to form a coalition government in a bid to end two months of political deadlock following March's inconclusive general election.
Mattarella must agree to the parties' candidate and ministerial team before they can seek approval for the new government in parliament.
CV controversy
Conte's official CV says that he "furthered his juridical studies" at Yale, New York University (NYU), Duquesne University in Pittsburgh, the Sorbonne and Cambridge, but some entries have been called into question.
NYU told AFP that their records did not "reflect Giuseppe Conte having been at the University as a student or having an appointment as a faculty member". It said he was granted permission to conduct research in the institution's law library between 2008 and 2014.
Cambridge University declined to give details about Conte.
Duquesne University told AFP he attended as part of an affiliation with Villa Nazareth, an exchange programme, and did legal research but "was not enrolled as a student".
Conte has yet to speak out publicly about the affair.
Five Star has defended him as its choice to head a cabinet in which M5S leader Luigi Di Maio and League chief Matteo Salvini are tipped to hold key posts.
"Conte is and absolutely remains Five Star and the League's prime ministerial candidate," state news agency ANSA quoted Di Maio as saying on Wednesday.
EU worries
Salvini meanwhile defended the coalition's eurosceptic candidate for economy minister, Paolo Savona.
Minister for industry between 1993-94, Savona was staunchly opposed to the signing of the Maastricht Treaty and considers the euro a "German cage".
"He is an expert with a solid background of studies but made the mistake of daring to say that the EU as it is isn't working," Salvini said.
"Why do you even bother to let us vote if when the people ask for radical change you tell us to be careful?"
European officials have expressed worry that Italy could trigger a new eurozone crisis by refusing to stick to public spending and debt targets set by Brussels.
EU Economic Affairs Commissioner Pierre Moscovici said on Wednesday that Italy must deliver a "credible" response on how it will reduce debt.
Italy has one of the eurozone's highest levels of government debt and lowest growth rates.
More than eight percent of its population lives in poverty, according to national statistics agency Istat.
Di Maio and Salvini's government programme pledges tax cuts and increased welfare spending to boost growth.
It also plans to speed up expulsions of illegal immigrants.
On Tuesday Salvini said there would have to be new elections if the coalition government was not given the go-ahead by Mattarella.
"Either we start and we begin the change or we may as well go back to the polls," he said in a live video on Facebook.
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