Labour has gone up three points since the last survey of its kind, while the Tories have dropped by two.
The figures come as a boost to Sir Keir Starmer and his Shadow Cabinet ahead of his first party conference as Labour leader.
The surge in public opinion relates both to the party as a whole and to Sir Keir’s performance at the helm.
His approval ratings are now ahead of Mr Johnson’s, the YouGov poll reveals, with some 38 per cent of respondents suggesting he looks like a “prime minister-in-waiting”.
Asked who would make the best Prime Minister, 34 per cent said Sir Keir, 30 per cent backed Mr Johnson, and 31 per cent said they did not know, The Times reported.
The poll also found that 35 per cent of people believe Sir Keir has changed the party for the better since taking over from Jeremy Corbyn, with just four per cent thinking he had changed it for the worse. A third of voters were undecided.
However, when it comes to Brexit, the public appear less confident.
Asked whether they would “trust” Labour to handle the UK’s departure from the European Union, 45 per cent of the poll's respondents said “no”, and only 29 per cent said “yes”.