It comes after Mr Cleverly said the plan, to deport asylum seekers arriving in Britain to Rwanda, was not the “be all and end all” of the government’s approach on small boats.
His approval rating among Tory members plummeted from 72 points last month to just 10,6 points in the latest poll.
“It’s fair to say that whatever you think of the new Home Secretary’s performance since he was appointed, pleasing Conservative activists seems to have been just about the last thing on his mind, for better or worse,” Conservative Home’s Paul Goodman and Henry Hill said.
The most popular cabinet minister among the Tory grassroots is Kemi Badenoch, with a commanding 63.4 points. The business secretary is seen as a potential successor to Mr Sunak should the Conservatives lose the next election.
She was backed in last summer’s leadership race by levelling up secretary Michael Gove,who at the time praised her “first class brain”.
Ms Badenoch, an anti-woke firebrand, came fourth in the contest, which eventually saw Liz Truss become prime minister.
Behind her in the Conservative Home survey are leader of the House of Commons Penny Mordaunt, veterans minister Johnny Mercer and the new so-called minister for common sense Esther McVey.
Lord Cameron, who made a sensational return to frontline politics when he took over as foreign secretary in last month’s reshuffle, is fourth from bottom in terms of popularity.
“Some of this will be Leave-ish sentiment, some an unease about his record on China, some perhaps a memory of the Greensill saga,” Conservative Home said.
He returned to the monthly survey on a rating of -4.9 points.
The prime minister’s ratings among Tory members has been wildly fluctuating for months, with Mr Sunak having been unable to dodge negative headlines.
Just hours after he met his key pledge to halve inflation this year, the government’s Rwanda scheme was ruled unlawful by the Supreme Court.
And just days after Mr Sunak and Jeremy Hunt’s tax-cutting autumn statement, figures showed net migration to the UK had soared to a record high.
The devastating rating for Mr Sunak among Conservative members come as his party sits 19 points behind Labour in the polls.