While a majority 53 per cent of voters still believe the 87-year-old monarch should rule for life, only a minority of 43 per cent said she should continue even if she was too ill to continue.
The Queen has been ruling since 1952.
Significantly, a solid 48 per cent believed she should stand down in those circumstances and allow the throne to pass to her heirs, the Sunday Times survey conducted by YouGov this week found.
The poll comes in the wake of an announcement earlier this week that the Queen's son and heir, Prince Charles, will take her place at this year's Commonwealth Heads of Government (CHOGM) meeting in Sri Lanka - a biennial conference she has not missed since 1973.
She has also cancelled a few other official engagements after being briefly hospitalised in March for a stomach bug.
Only 23 per cent still have doubts about his abilities, considerably lower than 37 per cent last year.
However, there still seems to be very little support for his wife Camilla, the Duchess of Cornwall.
Only 16 per cent believe she should become Queen, 46 per cent prefer the title of Princess Consort and 11 per cent believe she should have no title at all.
His wife Kate, the Duchess of Cambridge, was also rated a high 79 per cent in terms of her contribution since the royal wedding while colourful younger brother Prince Harry managed a decent 72 per cent.
YouGov questioned 1,945 adults on May 9 and 10 for its latest sample survey.
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