The department's website also said routine consular and all visa services at its embassy in Mexico City had been suspended.
"British nationals resident in or visiting Mexico may wish to consider whether they should remain in Mexico at this time," a statement said.
The French foreign ministry also strongly advised against visiting Mexico without an "imperative reason".
The advice followed an earlier warning from European Union's health chief.
The health commissioner, Androulla Vassiliou, said: "Personally, I'd try to avoid non-essential travel to the areas which are reported to be in the centre (of the crisis]. They should avoid travelling to Mexico or the United States unless it's very urgent for them."
Scottish airports are poised to initiate emergency procedures if the situation escalates.
It is likely that hundreds of Scots have been arriving at Scottish airports after flying from Mexico to London airports, or Charles de Gaulle in Paris, and catching connecting services.
Seasonal flights direct to Mexico from Glasgow International Airport begin on Friday.
Thomson will operate to Cancn every Friday, and Thomas Cook has a flight every Monday starting next week.
The first scheduled flight direct from Mexico to Glasgow will be a Thomas Cook service from Cancn arriving at 7:40am next Tuesday.
A spokeswoman for BAA Scottish Airports said before the Foreign Office's advice last night: "At the moment, it is business as usual. It shouldn't affect us at this stage. Our (direct] flights start at the end of the week."
She added: "We are ready to put a plan into action and will fully support any decision that the government makes."
This could involve passenger screening, with a doctor boarding planes arriving from Mexico to check if passengers had any symptoms of the disease.
A notice on the BAA Scottish Airports website advises passengers returning from an affected area to monitor their health for seven days. It says: "If you develop a feverish illness accompanied by one or more of a cough, sore throat, headache or muscle aches, you should stay at home and contact your GP by phone or seek advice from NHS Direct."
At Madrid's Airport, passengers arriving from Mexico had to fill out forms detailing where they had been, if they had felt any cold symptoms and leave a contact address and phone number. "Where we were in (Mexico], there was no real alarm, but we followed what was happening on the news, and we're a little bit worried," said Filomeno Ruiz, a Spaniard, just back from a week's holiday in Cancn.
Passengers were also given leaflets urging them to contact local health authorities if they noticed any symptoms in the next ten days.
In the airport's baggage arrival area, ground crews and police wore blue or white surgical masks. Some travellers took precautions even though they had not been to Mexico.
"Nobody has recommended it, but I've put the mask on out of precaution," said Briton Roger Holmes, who was travelling to Tunisia from Madrid. "I'm not afraid, but it costs nothing to be careful."
Governments in Asia leaped into action, scarred by potent memories of Sars and bird flu outbreaks.
Singapore, Thailand, Japan, Indonesia and the Philippines dusted off thermal scanners used during the 2003 Sars crisis and checked passengers arriving at airports from North America. South Korea and Indonesia introduced similar screening.
Cause for concern not alarm, says Obama
US PRESIDENT Barack Obama sought to reassure the public yesterday, as the country introduced new measures to combat the spread of swine flu.
Richard Besser, the head of the US Centres for Disease Control and Prevention said authorities were starting to undertake the so-called "passive screening", asking people about fever and illness and looking for people who are ill.
Mr Obama said the spread of swine flu was a cause for concern, but "not a cause of alarm".
He told a gathering of scientists that the administration was "closely monitoring" cases of swine flu. Forty cases have so far been confirmed in the US, but all have been mild and no patients have died.
Mr Besser said he was not reassured by the fact that so far in the US, no-one had died from the disease.
"From what we understand in Mexico, I think people need to be ready for the idea that we could see more severe cases in this country and possibly deaths," he said.
Mr Obama also said the American people could expect regular and frequent updates about what Washington was doing.
He said the swine flu threat demonstrated why the US could not allow itself to fall behind in scientific and medical research.
Yesterday, classes were cancelled at a New York school where a group of students, recently returned from a trip to Mexico, were among the confirmed cases of swine flu.
Students at St Francis School began falling ill after a group of friends returned on 19 April from holiday in Mexico.
School officials realised something was wrong last Thursday when about 75 students showed up at the nurse's office complaining of fevers, upset stomachs and achy bones.
HOLIDAY FIRMS' SHARES PLUNGE
SHARES in holiday firms and airlines tumbled yesterday as concerns escalated over the swine flu outbreak and its impact on the travel sector.
British Airways fell 7.8 per cent, while Thomas Cook and Thomson Holidays' parent firm TUI Travel were also suffering from the fear of a full-blown pandemic, slumping 4.4 and 3.3 per cent respectively on London's Stock Exchange.
Fellow blue chip stock Carnival, the cruise operator, was also hit, by a 6.8 per cent slide. However the wider London market proved resilient, with the FTSE 100 Index closing up 0.3 per cent.
It was a similar picture on Wall Street, with Starwood Hotels and Resorts Worldwide falling 9 per cent in early trading on the Dow Jones, and AMR, the parent company of American Airlines, sliding around 14 per cent. Oil prices also dropped to around $48 a barrel as a drop in demand was anticipated.