A day after Conservative MPs put them both into the final two, a poll of 730 Tory party members saw a “significant lead” for the foreign secretary over the former chancellor.
The YouGov survey, carried out after the two final candidates had been announced, on Wednesday and Thursday, put Truss ahead by 24 points.
The poll had Truss on 62% and Sunak on 38%, excluding those who do not know or would not vote.
The current size of the Conservative membership is unknown, but at the last leadership election in 2019 there were around 160,000 members, and insiders expect it to have grown, meaning the polling is not representative of the party.
A source in the Sunak campaign said: “In the last Conservative leadership contest, polls were well over 10% out. They can be wrong – and this is only day one of our focus turning to members.
“Rishi is going out to the membership and set out a positive vision for our country – and there is everything to play for.”
Sunak was the parliamentary party’s favourite, winning 137 votes to Truss’s 113 among Tory MPs.
But bookmakers placed the foreign secretary as the frontrunner, with early indications suggesting she is more popular with Tory members ahead of a summer of campaigning.
The pair battled to win the support of local politicians on Thursday when they took part in a private hustings for the Conservative Councillors’ Association.
They will then tour the UK to take part in 12 hustings for the Tory members who will vote for their next leader, with the result being announced on September 5.