The monthly cabimet league table for the ConservativeHome website gives the prime minister an approval rating of minus 25.4 – his lowest ever score.
Business and trade secretary Kemi Badenoch comes top of the poll on plus 63.4, ahead of Penny Mordaunt on 49.8 and Johnny Mercer on 45.4.
It means Sunak is currently the least popular member of the cabinet among the Conservative rank-and-file.
A total of 750 party members took part in the survey, which last month put Sunak ninth from bottom with an approval rating of plus 7.1. The month before, he was in eighth place on plus 26 points.
The survey took place in the wake of last week’s row with Greece over the Elgin Marbles, in which Sunak cancelled a meeting with the country’s prime minister for re-iterating his desire for the ancient sculpture to be returned to Athens from the UK.
Sunak is also under pressure from Tory MPs to set out how the government plans to respond to last month’s Supreme Court ruling that its Rwanda scheme is unlawful.
Meanwhile, the Tories remain well behind Labour in the opinion polls, despite the PM’s repeated attempts to revive the party’s fortunes.