Mr Johnson said: "We have a stockpile of drugs. Tamiflu deals with it once it's started and deals with it very effectively, we know from Mexico.
"Once we know what the strain is we will look to find a vaccine to prevent it and we have a pre-agreement for these vaccines to be produced as soon as we decide."
The World Health Organisation rates Britain as one of the two countries best prepared for an outbreak, along with France.
Scottish Health Secretary Nicola Sturgeon said the two Scottish patients who had visited Mexico both displayed mild flu-like symptoms but said there was no cause for concern.
Neither of the people had travelled in areas affected by swine flu outbreaks.
At least 71 people in Mexico are now thought to have died after contracting the new strain of flu being linked to an outbreak of swine flu in the US.
Fergus Walsh, BBC News medical correspondent
There are enough anti-viral drugs - such as Tamiflu - to treat one in two of the population. The government has placed orders to double that stockpile.
It can lessen the symptoms, and even prevent infection if given to close contacts of an infected person such as family members.
When Northamptonshire businessman Chris Clarke returned from Mexico on Saturday with mild flu-like symptoms he was told to stay at home with his family. He has not been seen by a doctor, but after calling NHS Direct doses of Tamiflu were dropped off for him and his two children.
That will be the typical response if other people return from Mexico and feel unwell. They are asked not to clog up hospitals. It is easy enough to arrange for one family - if 10,000 people need the same treatment it would be a different matter.
Justin McCracken, chief executive of the Health Protection Agency, told BBC News he thought there probably would be infections in the UK.
He said: "I think probably we should expect cases given the way this has spread across America. It is sensible that we plan in the assumption that there will be cases.
"We are already mobilising things in the UK in case the virus comes over here. I definitely think we have enough of the drugs."
Mr McCracken added: "I don't think at this stage there is any need to declare an emergency."
Professor Nigel Dimmock, a virologist from Warwick University, said it was unclear how much drug resistance this new strain may have.
He told BBC Radio 5 Live: "There is reason to be worried. The virus will travel and if it is, as seems, a new virus and people have no resistance to it, then there's nothing to stop it spreading from person to person and by various means around the world."
The Foreign and Commonwealth Office said people "should be aware" of the outbreak, but is not currently advising people against travelling to affected areas of Mexico and the US.
The advice said: "Cases have been reported in Mexico City, together with the states of Oaxaca, San Luis Potosi, Mexicali and Baja California."
Experts say it is impossible to know yet whether swine flu is the trigger for a pandemic - but within 72 hours it should become clear how far the virus has spread.
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