Currently, inoculated travellers must take a lateral flow test two days after they return to Britain. Anyone who tests positive must isolate for 10 days, although this can be cut short with a negative lateral flow test on days six and seven.
The announcement is reportedly set to be made on 26 January, when the prime minister is expected to lift plan B restrictions.
The Times, quoting a source close to transport secretary Grant Shapps, reported: "We are looking at removing all Covid tests for vaccinated travellers by the end of January, which is likely to coincide with the review of the plan B measures on January 26."