The YouGov survey of Tory party members for the broadcaster found 66 per cent are voting for foreign minister Truss, while 34 per cent back former finance minister Rishi Sunak.
The breakdown only includes those who have voted or plan to.
As the Conservatives currently have a majority in the UK parliament, either Truss or Sunak will become the country’s prime minister when current leader Boris Johnson leaves early next month.
The poll reported that more than half (57 per cent) had already cast their votes ahead of the ballot closing September 2.
Just over a quarter of those who still intend to cast a vote said they had not yet decided who to back.
The poll showed Sunak’s support had climbed by two per cent since a similar poll two weeks ago, but the “incremental improvement … would be nothing like enough to rescue the contest for him,” Sky News said.
The two leadership contenders have waged a bitter battle over recent weeks, featuring frequent hostile briefings and counter-briefings by their camps.
Underlining the Tory grassroots’ rift with the parliamentary party, the poll found that if Johnson were in the leadership contest, he would still win, with 46 percent saying they would vote for him.
He announced he would quit as leader of the ruling party earlier this summer after mass resignations among his cabinet members and MPs increasingly frustrated by his scandal-hit leadership.
The YouGov poll was carried out among 1,089 of the roughly 200,000 Tory party members.
The winner will be announced on September 5 and the victor will take charge the following day.
The candidates will hold the next of a dozen hustings during the summer-long contest in the northern English city of Manchester on Friday evening.
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