The likely end to the 14-month-old government would open the way for President Sergio Mattarella to begin consultations with political parties, with a range of options available.
"It is irresponsible to initiate a government crisis", Conte said adding that the move "shows personal and party interests".
"His decisions pose serious risks for this country."
Pressing for a new election as soon as possible, Salvini, who as interior minister has led a crackdown on migrants, said: "I don't fear Italians' judgment".
Salvini rejected Conte's comments, saying other parties were afraid of going to elections and losing their influence.
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Here's what to know about the Italian political crisis.
Salvini could be crowned prime minister with the League in coalition with the anti-immigration, anti-LGBT Fratelli d'Italia, and Forza Italia.
Conte's resignation was the culmination of a long waiting game.
By trying to trigger early elections, analysts believe Salvini hopes to capitalize on his party's popularity and become Italy's next prime minister. "Everything was about getting Salvini into the driver's seat before the budget", says Erik Jones, an Italy expert at the Johns Hopkins School of Advanced International Studies.
The premier scathingly quoted Salvini's own recent demands for an early election so he could gain "full powers" by grabbing the premiership.
Conte will address the Senate this afternoon at 3 pm but it is not clear yet whether he will resign immediately afterwards or choose to wait for the outcome of a no-confidence vote put forward by the Lega party, his far-right coalition partner.
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An agreement with PD remains problematic for the Five Star Movement, which considers the centre-left party and particularly former Prime Minister Matteo Renzi their "first enemies".
"Government crises are faced in Parliament, not in the public squares", Conte continued and slammed Salvini for various issues including his use of religious symbols, a critique often deployed by members of the Roman Catholic clergy hostile to Salvini's anti-mass migration policies.
"That is exactly what Salvini wanted to avoid", Jones tells TIME.
M5S leader Deputy Prime Minister Luigi di Maio said of his co-deputy "no one has lost esteem for Salvini more than me". Salvini's far-right party doubled its score from last year's 17.35% to 33.6%, while the Five Star Movement crumbled from 32.68% to 16.70%.
Negotiations between Italy's deeply divided parties would be hard, although some have mooted a new alliance between M5S and the centre-left Democratic Party (PD). Ever present on social media, Salvini has been taking his message to the beaches at the height of the summer holiday season, promising to cut taxes, to block migrants from Italian ports, and to complete a high-speed rail link to France.
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