Government agency Public Health England (PHE) said the total number of confirmed cases of the variant had more than doubled in the past week to 1,313 across the United Kingdom.
"We are anxious about it - it has been spreading," Johnson said, adding that there would be meetings later on Thursday to discuss what to do.
Asked about the possibility of imposing local lockdowns to contain the variant, Johnson said his government was "ruling nothing out."
Johnson's government is due to relax its nationwide lockdown in England on Monday May 17, with all major restrictions due to be scrapped just over a month later.
"At the moment I can see nothing that dissuades me from thinking we'll be able to go ahead on Monday and June 21 everywhere," he added.
Johnson said the government would outline more about "what the world will exactly look like from June 21" before the end of May.