Authorities are stepping up watchfulness at German airports
France says it will request an EU-wide ban on all flights to Mexico, source of the swine flu outbreak.
The move follows confirmation of new cases in Germany, Austria, Britain and Spain. Many, if not all, of those infected have recently been to Mexico.
French Health Minister Roselyne Bachelot said France would push for the flight ban at a meeting of EU health ministers on Thursday.
However World Health Organization (WHO) officials say a ban may be pointless.
"We will ask our European colleagues to consider the suspension of flights going to Mexico," Ms Bachelot said, after talks with President Nicolas Sarkozy.
She said there was no suggestion yet of stopping flights from Mexico back to the EU, which would leave Europeans stranded there or force them to find other ways home.
A UK Department for Transport spokesperson said any decision to suspend flights would be a matter for individual EU member states.
"We're taking advice from the WHO. If member states decide to suspend flights they will have to do it individually," the spokesperson told BBC News.
There has been some confusion over whether travel restrictions would be effective in preventing the spread of swine flu.
EU Health Commissioner Androulla Vassiliou on Monday advised people to "avoid non-essential travel" to the most affected areas, but was then forced to row back and state that this was personal, not official, advice.
Nevertheless, several countries, including the UK, have advised citizens against non-essential travel to Mexico.
This is despite the recommendation of WHO's assistant director-general Keiji Fukuda, who said trying to contain the virus by enforcing travel restrictions "is not a feasible option".
"Instituting travel bans would really not be very effective as the virus has already spread to several other countries," he said.
So far Mexico and the US are the only countries where victims of the virus have died.
The first US death was reported on Wednesday. In Mexico seven deaths have been confirmed - but scores are still under investigation.
On Wednesday Germany reported three confirmed cases, and Austria one, making them the third and fourth European nations hit by swine flu.
The German victims include two women, aged 22 and 37, and a man in his 30s. All had recently returned from Mexico.
German authorities say there is no need to panic - and that they have sufficient stocks of anti-flu drugs to protect the population, says the BBC's Tristana Moore in Berlin.
But the country is bracing itself for more cases and is increasing its watchfulness at airports like Hamburg and Frankfurt.
Meanwhile the UK says it has three more cases - making five in total.
UK Prime Minister Gordon Brown said the new British cases - two adults and a 12-year-old girl - were all in England. He said all three had mild symptoms and were responding well to treatment.
Spain's health ministry announced two more cases on Wednesday, making four in total, but said they were "not serious because they are responding well to treatment".
In Portugal, a three-year-old boy who arrived from the United States last week is the country's first suspected case, local media have reported.
On Tuesday EU Health Commissioner Androulla Vassiliou said suspected cases were also being investigated in the Republic of Ireland, Denmark, Italy, Sweden, Greece and the Czech Republic.
In Geneva, the World Health Organization is holding an emergency review of the swine flu outbreak, and said it might raise its pandemic alert level to phase five - the second highest - if it turned out that the disease was being spread in at least two countries, the Associated Press reported.