The company's estimated 12,500 employees cast their ballots from dawn on Thursday to midnight on Monday at Rome's Fiumicino airport and Malpensa and Linate in Milan, the source said.
Lengthy negotiations ended on Friday with a pre-accord approved by the powerful unions but which needs approval of workers.
The deal softens a tough restructuring package unveiled by management in mid-March, reducing job losses and salary cuts.
It would see 980 permanent job contracts axed instead of 1,338 and would trim air crew salaries by eight percent rather than between 24 and 30 percent.
In return, crew would agree to boost productivity by cutting the number of annual rest days to 108 from 120.
"No" campaigners are up in arms, saying employees have been bled dry in previous bids to revive the loss-making airline.
The company is de facto controlled by Etihad Airways, which acquired a 49 percent stake when it saved Alitalia from bankruptcy in 2014.
Nino Cortorillo of the Filt Cgil union said the deal was the best the unions could get after "long and difficult negotiations".
The pressure to find a solution is intense, with Alitalia's liquidity expected to run out this month without emergency funding, leaving its fleet grounded.
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Shareholders Etihad and the Italian banks Intesa Sanpaolo and UniCredit have said they will only inject new funds if the unions agree to the new collective labour agreement and cuts.
The Italian government, which acted as a mediator in the negotiations, warned on Tuesday that a "no" victory would be not only costly but potentially fatal for the company.
Alitalia has been hit hard by competition from low-cost companies and has been accumulating losses for years.