The February 19-20 survey by the Asahi newspaper also showed that 64 per cent want an early election to break the political stalemate that is bedevilling Japan as its economy slips deeper into recession.
The latest bad news for Aso comes ahead a visit to Washington, where he is scheduled to meet US President Barack Obama at the White House on Tuesday.
Aso, 68, suffered a severe blow this week when his finance minister quit after being forced to deny that he was drunk at a Group of Seven news conference in Rome.
The long-ruling conservative Liberal Democratic Party (LDP) tapped Aso last September in the hope he could lead them to victory in an election that must be held by October, but instead his ratings have plunged after a series of policy flip-flops and gaffes.
The Asahi survey showed support for Aso's cabinet at a mere 13 per cent, down one point from a February 7-8 poll. One TV survey has already put his rating at below 10 per cent.
Calls have emerged inside the LDP to replace Aso ahead of the election, which the party looks increasingly in danger of losing.
One possible timing is after parliament enacts the budget for the year from April 1. The annual budget is typically passed in March but could be delayed this year by the political deadlock.
But replacing Aso – Japan's third prime minister in less than two years – is not likely to rescue the LDP and its junior coalition partner from defeat at the polls, analysts say.
A loss would end the LDP's more than half a century of almost unbroken rule and usher in a government led by the opposition Democratic Party of Japan, which has pledged to reduce bureaucrats' control of policy, shrink social gaps and forge a diplomatic stance more independent of close ally Washington.
Forty-five per cent of voters said they saw Democratic Party leader Ichiro Ozawa, a former LDP heavyweight who bolted the party in 1993 and helped to oust it briefly, as suitable to be prime minister, compared to 19 per cent who preferred Aso, the Asahi poll showed.