The PM survived two confidence votes in parliament last week, but then lost his absolute majority in the Senate after centrist ally and former PM Matteo Renzi defected.
Parties are divided over spending in the coronavirus crisis, in which more than 85,000 Italians have died.
The country's president, Sergio Mattarella, will start consultations with the leaders of other parties on Wednesday, according to a statement, adding that PM Conte had been asked to stay on in a caretaker capacity
The centrist coalition government was plunged into crisis two weeks ago when former PM Matteo Renzi pulled his small, liberal Italia Viva party out of it. He said he would only return if PM Conte accepted a list of demands.
Renzi's main objection was to Conte's plans for spending €209bn ($254bn) of EU recovery funds - part of a €750bn EU rescue for the Covid crisis.
Renzi says EU funds should be invested in promising sectors like digital and green technologies, and wants MPs, rather than technocrats, to decide on the allocations. But he also wants more investment in the embattled health service.