The Ipsos research found 42% of people think Charles should step aside for his son William to take the throne.
This compares with 24% who think the prince should not stand aside for the duke, while 29% do not have strong feelings on the issue.
However, the survey also suggests that 48% of people in the UK still believe Charles will do a good job, compared with 19% who think he will do a bad job and 27% who think he will do neither a good nor bad job.
The research, which consisted of interviewing a representative sample of 2,055 British adults on March 24 and 25, also found the proportion of people who view Charles favourably has increased since 2018 to 43% – up 11 percentage points.
More than a third (36%) feel the same for his wife, the Duchess of Cornwall.
However Britons are much more likely to have a good opinion of the Queen, seen favourably by 69%, William, seen favourably by 64%, and the Duchess of Cambridge, seen favourably by 60%.
Yet opinion of Charles is higher than the Duke and Duchess of Sussex, whose popularity has plummetted over the past four years.
Ipsos found that only 30% see Harry in a favourable light – down 35 percentage points from 2018 – while just 24% have a good opinion of Meghan, down 16 points.
Meanwhile, seven in 10 (69%) now have an unfavourable opinion of the Duke of York, in the wake of the sex scandal case involving Virginia Giuffre, Ghislaine Maxwell, and Jeffrey Epstein.
The research also looked at where Britons stand on abolishing the monarchy, and on other members of the royal family, including Andrew and the Duke and Duchess of Sussex.
More than four in 10 people (44%) think abolishing the monarchy would be worse for Britain – twice as many as the 22% who think it would be better.
Gideon Skinner, head of political research at Ipsos, said: “At the moment only a minority of people believe Britain would be better off without the monarchy, and the popularity of the Queen and the Duke and Duchess of Cambridge remains strong.
“Looking to the future, half of Britons have faith in Prince Charles as a future king and more people view him in a positive light than four years ago.
“However, with the popularity of Prince William not far behind that of the Queen, Britons also wonder whether Prince Charles should step aside in favour of his eldest son.
“But their actions now are still important for the longer-term future of the royal family.”