The Deltapoll survey of the 57 constituencies the Tories gained in the 2019 General Election, most of which could be considered part of the Red Wall, found that 38% of voters agreed Sir Keir would make the best PM, while 33% backed Mr Johnson.
The poll, for the Mail on Sunday (MoS), also put Labour in the lead when it came to the voting intention in those seats (49% vs 33%).
Meanwhile, it suggested Labour was ahead by five points nationwide (40% vs 35%).
The newspaper said if the results were repeated in a General Election it could cost the Conservatives more than 100 seats.
This would be enough to win Sir Keir a spot in Downing Street, the MoS said, but leave him lacking a clear majority.
Voters in the seats gained by the Tories also preferred Sir Keir and shadow chancellor Rachel Reeves for their top team (40%), to Mr Johnson and Rishi Sunak (33%).
Asked if Mr Johnson was doing well in general, 34% agreed, while 62% disagreed – giving the PM a net rating of minus 28.
Sir Keir also received a negative score, but slightly higher, at minus 6.
The picture was even worse for the Prime Minister when voters in the 57 seats were quizzed on the pandemic. Only 32% thought he was doing well on Covid, while 62% disagreed.
Meanwhile, just 16% believed Mr Johnson had obeyed the rules, compared to 72% who did not.
A majority (65%) said the PM should resign if he was found to have broken the law over Covid restrictions.
Mr Sunak, on the other hand, proved more popular with these voters when it came to his pandemic performance – with 45% agreeing he had done well, compared to 40% who disagreed.
Looking forward to this time next year, just 24% of voters in the seats gained by the Tories thought Mr Johnson would still be in the top job. A majority (58%) thought he would be replaced.
And nearly three in four (74%) said they did not trust the PM to tell the truth.
The MoS said Deltapoll came up with its figure of 57 constituencies based on the 58 taken from other parties by the Tories in 2019, minus the seat of former Speaker John Bercow who stood down.
Writing in the MoS, a Tory MP who gained his seat from Labour in the 2019 election said some first-time Conservative voters were starting to have doubts about the party.
Lee Anderson, MP for Ashfield, said the Government’s achievements were being soured by “the huge rises in the cost of living coming down the track, through higher energy bills, which my voters care far more about than the platitudes spouted about the ‘green agenda’ by the wealthy elite who flew into the Cop26 summit in private planes”.