The US government has declared a public health emergency and advised against non-essential travel to Mexico, a call echoed by Androulla Vassiliou, the European Union's health chief.
Mexico on Monday also closed all schools across the country until May 6 as a precaution, officials said. Schools had already been closed in Mexico City and in five of Mexico's 32 states.
The World Health Organisation (WHO) is meeting in Geneva, Switzerland, to decide whether it should raise its pandemic alert level over the flu outbreak.
Barack Obama, the US president, said that officials are closely monitoring the cases of swine flu in the country.
"This is obviously a cause for concern and requires a heightened state of alert. But it is not a cause for alarm," he said on Monday.
The new H1N1 virus, apparently a mix of human, pig and bird viruses, has so far not been the apparent cause of any deaths outside Mexico.
People are also being tested for suspected cases in France, Italy, Sweden and Switzerland.
Other suspected cases are being investigated in Brazil, New Zealand and Israel.
Russia, Hong Kong and Taiwan have said travellers returning from flu-affected areas who displayed symptoms of fever would be quarantined.
Several countries have also banned imports of pork and pork products from Mexico and the three US states reporting cases of swine flu, although scientists say the disease would not be contracted by eating pork meat.
|Mexico City residents have been given surgical masks and urged to stay home [AFP]
New Zealand, Israel and several other countries have implemented new security measures at airports and are putting sick visitors under observation.
WHO currently has an alert level of three for the virus, which means the animal flu does not spread well between humans.
If raised to four or five, it will mean the virus is becoming adept at spreading between humans, and could also mean further travel advisories, restricted trade and cancelled sporting and entertainment events.
On Sunday, Felipe Calderon, the Mexican president, urged citizens to wash their hands, stay home and avoid crowded places.
The government has declared an emergency in the country, giving it powers to isolate people, enter homes or workplaces and regulate air, sea and land transportation to try to stop further infection.
Peter Cordingley, WHO's regional representative in Manila, the Philippines, said the organisation believes the virus cannot be contained at a local level.
Deaths: At least 20 confirmed, 129 now suspected in Mexico
Sickened: 1,614 suspected or confirmed in Mexico, 40 confirmed in US, six confirmed in Canada; two in UK, suspected cases in New Zealand, Spain, Brazil, France and Israel
Safety measures in Mexico: Schools closed until May 6, surgical masks given to train passengers, public events cancelled, schools and public venues closed and church services postponed
Safety measures worldwide: Airports screening travellers from Mexico for flu symptoms
Hong Kong and South Korea warn against travel to Mexico
Some countries increasing screening of pigs and pork imports or banning them outright
"We've seen already that it has spread to the United States. Whether it can be contained globally we don't know because we don't know at this stage whether it is going to move globally, although there are good reasons to believe that there is a potential for a pandemic," he told Al Jazeera.
"Even with avian influenza, when it seemed to stall and it lost the interest of the public, we continued to talk about the pandemic potential. What we are seeing now is a situation that has accelerated far, far faster than whatever happened with avian influenza. We are seriously concerned."
Richard Besser, the acting head of the US Centres for Disease Control (CDC), told a White House press conference that "we expect to see more cases of swine flu".
"We're responding aggressively to try and learn more about this outbreak," he said.
Margaret Chan, the WHO chief, warned on Saturday that the "virus has clearly a pandemic potential".
Keiji Fukuda, the acting WHO assistant-general for health, security and the environment, said the virus could mutate into a "more dangerous strain".
But Fukuda said that if there were such an escalation, the world was better prepared now than it has been in the past.
After the outbreak of Severe Acute Respiratory Syndrome (Sars) in 2003, which killed nearly 800 people, and amid regular reported cases of bird flu, countries have stepped up preparations against a pandemic, including stockpiling anti-viral drugs.