Austria, Croatia and Switzerland said the cases involved people who had been to Italy, as did Algeria in Africa.
The first positive virus test has been recorded in Latin America - a Brazilian resident just returned from Italy.
Italy has in recent days become Europe's worst-affected country, with more than 300 cases and 11 deaths.
But its neighbours have decided closing borders would be "disproportionate".
Health ministers from France, Germany, Italy and the EU Commission committed to keeping frontiers open at a meeting on Tuesday as new cases of the virus emerged throughout Europe and in central and southern Italy.
"We're talking about a virus that doesn't respect borders," said Italian Health Minister Roberto Speranza.
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Iran's deputy health minister and an MP have both tested positive for the new coronavirus, as it struggles to contain an outbreak that has killed 15.
The country is one of three hot-spots outside China causing concern that the virus could be becoming a pandemic.
The deputy minister, Iraj Harirchi, on Monday denied covering up the scale of the outbreak. He appeared in physical discomfort as he spoke to reporters.
Iran has reported 95 cases, but the actual number is thought to be higher.
The director general of the World Health Organization (WHO) has said the sudden increase in cases in countries outside China is "deeply concerning".
US Secretary of State Mike Pompeo said on Tuesday that Washington was concerned that Iran may have concealed "vital details" about its outbreak and urged all nations to "tell the truth about the coronavirus".
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