"We are continuing to ask Berlusconi to make a gesture of responsibility and help us give this country a government", said senior League politician Giancarlo Giorgetti, indicating he wanted Forza Italia to agree to sit out of a government deal.
Without their support, Mr Mattarella is unlikely to get his proposal through parliament, said Luigi Ferrata, public affairs account director at Community Group, which is a political consultancy based in Rome.
The payback for Berlusconi was unclear, amid speculation that he would seek guarantees that the new government would not push laws hampering his business empire, or deepen his legal woes.
Although Salvini and Di Maio are not entirely out of the running, others mentioned in the Italian media as potential prime ministers include Enrico Giovannini, a former chief statistician at the Organisation for Economic Co-operation and Development, and Giulia Bongiorno, a lawyer who made a name for herself after her defence of Giulio Andreotti, a former prime minister found guilty in 2002 of complicity in a mafia-linked hit against a journalist. Di Maio could serve as foreign minister, and Salvini as interior minister, the paper said.
Berlusconi, 81, had been the stumbling block in earlier negotiations between the League and M5S, with the four-time ex-prime minister seen as a potent symbol of the corruption they have campaigned against.
Investors were also selling Italian government debt, sending yields higher.
"We still have to work on the programme, on dates, on the team and the things that need doing", Salvini said. "Either we reach a deal quickly, or we vote".
In a joint statement after meeting on Thursday, Mr Di Maio and Mr Salvini described a "positive climate to define the government's agenda and priorities" and said "technical" meetings among staff would begin in the afternoon.
Mattarella called on the bickering parties to show "responsibility" and support his now unnamed government, adding that a new vote would have to be held if such an administration failed to find the necessary backing in parliament.
"A political government is now more probable, and a mild negative market reaction is likely", said Vincenzo Longo, an analyst at IG Markets. Late on Wednesday, Berlusconi accepted a demand from 5-Star that his Forza Italia party take no part in the next government. "This coalition was the most feared by markets even before the vote, given their anti-establishment views, and market impact has been limited by their less extreme views on the euro and Europe since their campaign, but the impact may only emerge in the longer term".
"It certainly won't be us who imposes vetoes", Berlusconi said in a statement, adding that although he would not support this new coalition in parliament, his partnership with the League would still continue at a local level. The government would then dissolve at the end of the year ahead of elections planned for early 2019. If the premier loses the confidence vote, Mattarella will dissolve parliament.