Those travelling to the US will be required to get a test in the 24 hours before they depart, regardless of vaccination status.
The measures will come into force from next week.
The requirement for people to wear face coverings on planes, trains and buses will also be extended until mid-March.
Cases of the Omicron variant have been detected in the US: in California, Colorado, Minnesota, and New York, and in Hawaii, where authorities say that person had no recent travel history.
Experts stress there is still much to be known about the Omicron variant, including whether it is more transmissible or better at evading vaccines.
From December 1, all arrivals in Portugal must show evidence of a negative Covid-19 test, regardless of vaccination status.
For entry to Switzerland, Britons must show a negative PCR test and quarantine for 10 days.
Spain will only be accepting fully vaccinated British tourists aged 12 and over from next month.
From Saturday 4 December, France will require all visitors from countries outside the EU to provide a negative Covid test result on entry to the country – regardless of vaccination status.
England has no requirement for pre-departure tests, but all arrivals must take a PCR test on or before day two and quarantine at home until receiving a negative result.
It comes as World Health Organisation (WHO) experts say Omicron cases are increasing daily worldwide and countries should prepare for potential surges.
The WHO’s regional director for the western Pacific, Dr Takeshi Kasai, says the geographic distribution of the new Omicron variant is “likely already wider than currently reported”.
Meanwhile, Dr Babatunde Olowokure, WHO regional emergency director, says that blanket travel bans can only delay the entry of the variant into countries and that countries should re-think their approach to limiting the spread.
The WHO emphasized the importance of mask wearing, social distancing, hand hygiene, avoiding large gatherings, contact tracing and getting vaccinated.