The UN has warned that the virus has the potential to become a pandemic
The first cases of swine flu have been confirmed in Europe.
Two British people admitted to hospital in Scotland after returning from a holiday in Mexico have been confirmed as having the virus.
In Spain, a man has tested positive for swine flu and 17 other people are under investigation, officials said.
EU health ministers are to meet on Thursday to discuss the outbreak, which health officials suspect has killed about 150 people in Mexico.
The World Health Organization (WHO) raised its alert level late on Tuesday from three to four - two steps short of declaring a full pandemic.
WHO Assistant Director General Dr Keiji Fukuda said it signalled a "significant step towards pandemic influenza", but added "we are not there yet".
The EU's Health Commissioner, Androulla Vassiliou, has advised against non-essential travel to any affected areas.
Besides Mexico, the UK and Spain, there have also been confirmed cases in the US and Canada. Suspected cases are being investigated in Brazil, Israel, Australia and New Zealand.
"Personally, I'd try to avoid non-essential travel to the areas which are reported to be in the centre of the cluster," Ms Vassiliou said.
The precaution would "minimise the personal risk and to reduce the potential risk to spread the infection to other people", she added.
Germany's largest tour operator, TUI, suspended all trips to Mexico City as a precaution - though holidays to other parts of Mexico would continue as normal.
Scottish Health Secretary Nicola Sturgeon said both UK patients were recovering well in a Lanarkshire hospital from mild cases of the virus.
Swine flu is a respiratory disease thought to spread through coughing and sneezing
Symptoms mimic those of normal flu - but in Mexico more than 100 people have died
Good hygiene like using a tissue and washing hands thoroughly can help reduce transmission
"The threat to the public remains low," she said.
"The precautionary actions we have taken over the last two days have been important in allowing us to respond appropriately and give us the best prospect of disrupting the spread of the virus."
There are 14 other possible cases currently being investigated in the UK, though all the people involved are well enough to be managed in the community, British Health Secretary Alan Johnson said.
In Madrid, Spanish Health Minister Trinidad Jimenez confirmed that a male patient in Almansa in the south-eastern province of Albacete had tested positive for the virus.
The 23-year-old man had arrived home from Mexico with a fever last Wednesday, and was isolated in a hospital ward on Saturday night, she added.
Ms Jimenez said his condition was not serious.
At least 17 further suspected cases are under investigation in Spain, in locations ranging from Madrid to Mallorca, Barcelona to the Basque Country.
The patients have all recently returned from Mexico. None of the cases is thought to be life threatening.
Robert Madelin said it was clear that a 'novel' flu virus was now being confronted
Four people tested for the virus in France have been given the all-clear by doctors, French Health Minister Roselyne Bachelot has said.
The WHO has said the swine flu virus could be capable of mutating into a more dangerous strain, but officials say they need more information on the virus to determine the threat it poses.
They are advising all countries to be vigilant for seasonally unusual flu or pneumonia-like symptoms among their populations - particularly among young adults, a characteristic of past pandemics.
The H1N1 virus is the same strain that causes seasonal flu outbreaks in humans, but the newly detected version contains genetic material from versions of flu which usually affect pigs and birds.
There is currently no vaccine for the new strain, but severe cases can be treated with anti-viral medication.
Earlier, the European Commission convened an emergency meeting of the health ministers from all the EU's 27 member states on Thursday to discuss growing concern about the outbreak.
SPREAD OF VIRUS
Mexico: 20 confirmed cases (Mexico puts suspected deaths at 149)
United States: 40 confirmed cases
Canada: 6 confirmed cases
Spain: 1 confirmed case
UK: Scotland says tests confirm 2 cases
Israel, Brazil, Guatemala, Peru, Australia and New Zealand: Suspected cases being tested
"The commission is following the situation concerning the swine influenza very closely," said Commission President Jose Manuel Barroso.
"We will continue to assess the information we are getting from the experts, evaluate the potential danger and decide together with member states on the measures to take."
Further details about Thursday's meeting have to be worked out by the Czech Republic, which currently holds the EU presidency.
The commission's director-general for health and consumer protection, Robert Madelin, said close co-ordination between EU members was the most effective way to tackle any pandemic.
He said the EU was "putting in place plans agreed in the last five years, ensuring that the level of risk management is appropriate to the risk - using our 'war-games' experiences".