Early evidence suggests Omicron has a higher re-infection risk.
The World Health Organization (WHO) said on Friday that the new variant was "of concern".
Several cases have now been identified in Europe - two in the UK, two in Germany, one in Belgium and another one in Italy, while a suspected case was found in the Czech Republic.
Israel, where the new variant has been confirmed, has decided to ban all foreigners from entering the country from midnight Sunday.
The measure will last for 14 days, according to the Times of Israel.
Omicron cases have also been detected in Botswana, Hong Kong and Israel.
Hundreds of passengers arriving in the Netherlands from South Africa are being tested for the new variant.
Some 61 people on two KLM flights were positive for Covid-19 and have been quarantined at a hotel near Amsterdam's Schiphol airport while they have further tests, Dutch officials said.
The Netherlands is currently struggling with a record-breaking surge in cases. An extended partial lockdown comes into force there on Sunday evening.
A statement by the South African foreign ministry on Saturday strongly criticised the travel bans.
"Excellent science should be applauded and not punished," it said.
The bans were "akin to punishing South Africa for its advanced genomic sequencing and the ability to detect new variants quicker".
The statement added that the reaction had been completely different when new variants were discovered elsewhere in the world.
An African Union official told the BBC developed countries were to blame for the emergence of the variant.
"What is going on right now is inevitable, it's a result of the world's failure to vaccinate in an equitable, urgent and speedy manner. It is as a result of hoarding [of vaccines] by high-income countries of the world, and quite frankly it is unacceptable," said AU vaccine delivery alliance co-chair Ayoade Alakija.
"These travel bans are based in politics, and not in science. It is wrong... Why are we locking away Africa when this virus is already on three continents?"