The number of Tory backers saying the former foreign secretary lacks the qualities needed by an occupant of Number 10 has risen seven points to 58 per cent since September, the Ipsos MORI survey states.
Conservative supporters believing Mr Johnson has the right prime ministerial abilities dropped from 39 per cent to 32 per cent in the same period.
Other potential Tory leadership contenders receive similar approval ratings, with 32 per cent of Conservative backers saying foreign secretary Jeremy Hunt and home secretary Sajid Javid have what it takes to be PM.
Work and pensions secretary Amber Rudd is just behind on 31 per cent.
But the possible leadership rivals score smaller negative ratings than the ex-foreign secretary.
Mr Javid is seen as lacking what it takes to be prime minister by 40 per cent of Tory supporters, while 38 per cent take the same view of Ms Rudd and Mr Hunt gets a 36 per cent score.
Environment secretary Michael Gove is seen as good prime minister material by 30 per cent of Tory backers, with 45 per cent disagreeing.
Ex-Brexit secretary Dominic Raab is rated as having what it takes by 20 per cent, with 44 per cent taking the opposite view.
Three in five Conservative supporters agree Theresa May has the qualities needed to be a good prime minister, while 30 per cent disagree.
Among all voters, the figures are almost reversed, with 30 per cent saying Mrs May has the qualities needed by a good PM and 57 per cent taking the opposite view.
The situation is worse for Labour leader Jeremy Corbyn, with 21 per cent of all voters saying he has what it takes to be in Downing Street, while 67 per cent disagree.
Views are split on when Mrs May should stand down as prime minister, with 36 per cent of all voters saying the Tories should change their leader as soon as possible, a rise of seven points since September.
Just 30 per cent of voters think the Tories should elect a new leader once Britain has left the EU, but before the next general election.
And 24 per cent believe the Conservatives should get a new leader after the next general election.
Of Conservative backers, 17 per cent think the party leader should be changed as soon as possible, with 39 per cent saying it should happen after Brexit, but before the next general election.
Ipsos MORI interviewed 1,050 people aged 18 and above by telephone between 15-19 March for the poll.