The cultural ministry launched the international search in May for directors to lead sites including Rome's National Etruscan Museum and Trieste's Historical Museum of the Castle of Miramare.
Out of the 309 who have applied, 26 are foreign or have dual citizenship, the ministry said on Monday, while 201 are women and 189 are men.
A team of experts will now make a preliminary shortlist of no more than 10 candidates per job, with a final decision on the appointments being made at the end of the year.
“The museum revolution continues,” Culture Minister Dario Franceschini said.
The other sites include the Ancient Roman ruins of Ostia Antica, Hadrian's Villa and Villa d'Este in Tivoli, the Pilotta monumental complex in Parma, Herculaneum, the Phlegrean Fields archaeological park and the Museum of Roman Civilisation.
The call for applicants is the second phase of a government plan to make Italy's museums – which boast some of the greatest archaeological remains and most iconic pieces of art in the western world – more competitive.
An international recruitment drive in 2015 saw the government appoint seven foreign directors at some of the country's biggest museums in a bid to boost visitor numbers.
Despite the cultural riches Italy owns, not one Italian museum is in the top ten globally in terms of visitors.
Florence's Uffizi art gallery pulls in just 1.5 million visitors each year, a fraction of the 8.5 million who visit the Louvre gallery in Paris each year.
The four-year roles pay €78,000 a year, plus bonuses.