Fiat has said that it has made 20,000 fewer cars over the last three weeks due to the protests.
Fiat chief executive Sergio Marchionne is due to meet Mr Monti later.
A Fiat spokeswoman told the BBC that its plants - four in the south of Italy and one in Turin - had closed on Friday.
The closure includes its Pomigliano plant, near Naples, where it is making the new Panda.
Plant closures in the past weeks had stopped production and it predicted a 10% loss of its Italian market share for this month if drivers don't resume work in the coming days.
Separately, it was revealed this week that Mr Marchionne received 14.5m euros (£12m) in compensation - 2.5m euros in cash and the rest in shares.
The Fiat Group is the largest private taxpayer in Italy and also controls US carmaker Chrysler.
Since taking over as chief executive in 2004, Mr Marchionne has been reorganising the company's car plants in Italy.
Despite resistance from unions, more flexible work practices have been introduced.
But according to the company, Fiat's Italian plants are running at just 50% of their potential capacity, while plants in the US and Poland are working at full speed.
Mr Monti leads a government of technocrats tasked with leading Italy out of its debt crisis, and has pushed through a package of spending cuts, tax rises and pension reforms.
Mr Monti took over from Silvio Berlusconi last year in response to the crisis, and says that without the measures Italy would face economic disaster like that which has affected Greece.