Health ministers from Italy's neighbouring countries say closing frontiers would be a "disproportionate and ineffective" measure.
Italy's European neighbours pledged to keep borders open Tuesday despite the new coronavirus spreading down the country to Tuscany and Sicily and a surge in the number of infected people.
Prime Minister Giuseppe Conte has blamed poor management in a hospital in the country's north for the outbreak, with the death toll in Italy rising to 11 and infections to 322, the largest number of people infected in Europe.
All of those who have died so far in Italy were either elderly or had pre-existing medical conditions.
Even as numbers of infections continue to rise, health ministers from Italy's neighbours - meeting in Rome along with the EU's health commissioner - said closing frontiers would be a "disproportionate and ineffective" measure.
"We're talking about a virus that doesn't respect borders," said Italy's Health Minister Roberto Speranza.
His German counterpart, Jens Spahn, who was also at the Rome meeting, said they were "taking the situation very, very seriously.
"The coronavirus has reached Europe for the first time in a situation where we don't understand every chain of infection and they can't be connected directly to China.
"This means we have a new situation to deal with. I have said it could get worse before it gets better and this assessment still stands," he added.
Virus spreading south
Tuscany reported its first two cases, including one in the tourist destination of Florence, while three emerged in Sicily - including a husband and wife from the worst-hit Lombardy region, where 240 people have tested positive.
The Liguria region, known as the Italian Riviera, also reported its first case.
Italy's 11th death, a 76-year old woman in the Veneto region, was announced late Tuesday by the region, according to multiple media sources. The next official update of the toll was expected Wednesday.
Hundreds of people were confined to their rooms in a Tenerife hotel in Spain after an Italian tourist was hospitalised with suspected coronavirus, health officials in the Canary Islands said.
Croatia confirmed the first case in the Balkans region after a young man recently returned from Italy - which lies across the Adriatic from Croatia - was found to have become infected.
Austria also saw its first two cases confirmed on Tuesday in the Tyrol province, which borders Italy.
One was an Italian receptionist working at a hotel in the Alpine city of Innsbruck, which was consequently put under lockdown.
Switzerland also reported its first case.
France reported two new cases, one a Frenchman recently returned from Lombardy.
While borders remain open, several governments have announced additional measures for incoming travellers, in particular from the two northern regions of Lombardy and Veneto.
They range from medical screening to special gates at airports and recommendations to self-isolate.
Conte insisted that Italy's health protocols were "among the most rigorous".
But on Tuesday evening, France's junior transport minister recommended people avoid travel to those parts of Italy hit by the outbreak.
Wide-ranging measures to halt the spread of the virus have affected tens of millions of people in the north of Italy, with schools closed and cultural and sporting events cancelled.
Several upcoming football matches in Italian Serie A and the Europa League will be played behind closed doors.
Ireland's government recommended the cancellation of next month's Six Nations rugby match in Dublin with Italy.
Production of the latest "Mission: Impossible" film starring Tom Cruise in Venice has also been stopped.
The main centre of infection in Italy has been the town of Codogno, a town of some 15,000 people around 60 kilometres to the south of Milan.
Codogno and several other towns in northern Italy have been put under isolation.
The 38-year-old man dubbed "Patient One" by Italian media was admitted to hospital last Wednesday in Codogno, and it is thought a large number of the cases in the worst-hit region of Lombardy can be traced back to him.
His heavily pregnant wife, several doctors, staff and patients at the hospital are thought to have caught the virus from him.