An exclusive poll by Ipsos MORI shows both leadership hopefuls up four points since March to be seen by 36 per cent as having what it takes to be a good Prime Minister.
But just over half of Conservative backers do not think Mr Johnson has the qualities needed.
The poll is the first test of Tory opinion since Rory Stewart joined the Cabinet and Mrs May was pressed to step down by the 1922 Committee.
The state of the race is:
- Home Secretary Mr Javid is the first candidate without a net negative rating. His score has been rising since July 2018.
- Mr Johnson attracts fewest “don’t knows” but also the highest negative reaction. The proportion who think he would not be a good PM is down from 68 per cent in 2016 to 53.
- Former Brexit Secretary Dominic Raab is on 22 per cent, up two. The percent-age who think he would not be good is down sharply since March from 44 to 29.
- Amber Rudd is broadly the same, with 30 per cent saying she would be good, while 38 per cent disagree.
- Foreign Secretary Jeremy Hunt has 29 per cent saying he would be a good PM (down three points) and 41 per cent disagreeing (up five).
- Environment Secretary Michael Gove’s ratings dip to 26 per cent favourable (down four) and 48 unfavourable (up three).
- Defence Secretary Penny Mordaunt has a 21 per cent positive score, and 28 per cent negative.
- International Aid Secretary Rory Stewart is least well known, with 17 per cent saying he would be good and 24 not.