Ireland, Germany, France, Italy, the Netherlands and Belgium are all halting flights and travel. The measures vary and are initially generally short-term.
An EU meeting will be held on Monday morning to discuss a more co-ordinated response.
The new variant has spread quickly in London and south-east England.
Prime Minister Boris Johnson on Saturday introduced a new tier four level of restrictions for those areas, scrapping a planned relaxation of rules over the Christmas period for millions of people.
Top health officials said that there was no evidence the new variant was more deadly, or would react differently to vaccines, but it was proving to be up to 70% more transmissible.