Health Minister Beatrice Lorenzin said the initial tests had allowed authorities to breathe a "sigh of relief" but confirmed that tests were continuing to completely rule out any causal link between the vaccine and the fatalities.
Aifa said the 19 reported deaths followed vaccinations from a total of eight batches of the drug made by the Swiss pharmaceuticals giant.
The two batches tested so far were linked last week to the deaths of two women, aged 79 and 87, and a 68-year-old man, all in the south of the country.
Aifa and the Higher Health Institute said the additional cases notified to them were probably the result of media coverage of the first cases and urged the elderly in particular to ensure they got protection against seasonal flu, which kills 8,000 people a year in Italy.
In a bid to reassure the public, Aifa Director General Luca Pani had the vaccine administered to himself on television station RAI Uno on Monday.
Novartis has said there is no evidence to link the deaths to the vaccine, which has been in use since 1997 and of which 65 million doses have been given to people around the world.