However, there remains huge uncertainty around the numbers - and vaccines are still expected to work.
The data comes from mathematicians comparing death rates in people infected with either the new or the old versions of the virus.
The new more infectious variant has already spread widely across the UK.
Mr Johnson told a Downing Street briefing: "In addition to spreading more quickly, it also now appears that there is some evidence that the new variant - the variant that was first identified in London and the south east - may be associated with a higher degree of mortality.
"It's largely the impact of this new variant that means the NHS is under such intense pressure."
Public Health England, Imperial College London, the London School of Hygiene and Tropical Medicine and the University of Exeter have each been trying to assess how deadly the new variant is.
Their evidence has been assessed by scientists on the New and Emerging Respiratory Virus Threats Advisory Group (Nervtag).
The group concluded there was a "realistic possibility" that the virus had become more deadly, but this is far from certain.
Sir Patrick Vallance, the government's chief scientific adviser, described the data so far as "not yet strong".
He said: "I want to stress that there's a lot of uncertainty around these numbers and we need more work to get a precise handle on it, but it obviously is a concern that this has an increase in mortality as well as an increase in transmissibility."
