That has left President Sergio Mattarella with the crucial task of sounding out political leaders before deciding whether to declare the end of the national legislature 3½ years early.
Salvini's rival in the outgoing government, Deputy Premier Luigi Di Maio of the populist 5-Star Movement, was to meet with Mattarella later Thursday. The outcome of that meeting was expected to indicate if the 5-Stars, Parliament's biggest party, would entertain the Democrats' overture to counter Salvini.
Nicola Zingaretti, who leads the center-left Democrats, stressed after meeting with Mattarella at the presidential Quirinal Palace that any new coalition would have to nail down Parliament's durable, broad backing.
"Not a government at any cost," Zingaretti told reporters at the palace. "We need a government that changes direction, an alternative to the right, with a new, solid program, a broad base in Parliament which gives back hope to Italians."
Zingaretti acknowledged immediately that it was a gamble but contended it was "useful" to try. If the Democrats and the 5-Stars fail to work together, "the natural way out of the crisis is new, early elections," Zingaretti said.
Creating a viable replacement for Conte's 14-month-old coalition, an unnatural alliance of Salvini's anti-migrant League and the anti-establishment 5-Stars, could prove a herculean task.
Both the Democrats and the 5-Stars have been weakened by infighting — and they failed to agree to a coalition deal after the 2018 election that ultimately brought Conte's now-caretaker government to power.
Zingaretti said any new government formed now would have to pledge to protect the "pro-European vocation" of Italy. The 5-Stars, however, frequently depict European Union policies as infringing on Italy's autonomy.
Mammoth state spending under Conte's tenure, reflecting populist promises to voters by both the 5-Stars and League, means whoever governs Italy for the rest of this year must slash tens of billions of euros from the proposed 2020 budget to avoid triggering higher sales taxes and other unpopular measures.
Salvini with his "Italians first" agenda has openly challenged the EU's financial rules for the 19 nations including Italy who use the shared euro currency.
The proposal for a possible new government to stave off a new election was promptly rejected by former conservative Premier Silvio Berlusconi, who met with Mattarella after the Democrats. The center-right leader warned against "an improvised majority that exists only in Parliament and not in the country."
The media mogul described his Forza Italia party as the nation's best guarantee that, should it return to power in a right-wing-led government, it would guard against abandoning the euro currency or other pro-European policies.
Berlusconi also has a personal reason to welcome a new election. He is now again eligible to hold public office, after a tax fraud conviction forced him to surrender his Senate seat in the previous legislature.
Frances D'Emilio is on twitter at www.twitter.com/fdemilio .