Over half of people surveyed in a Northern newspaper poll believe that Peter Robinson should resign as First Minister and DUP leader because of revelations about his personal life.
The poll also found that a third are in favour of Gerry Adams resigning as Sinn Féin president following allegations surrounding his brother.
The survey was commissioned by the Belfast Telegraph and was carried out over a five-day period after last week's Hillsborough agreement between the DUP and Sinn Féin.
Respondents were asked how they rated the performance of the North's main political parties on their handling of the issue of devolution of policing and justice.
Sinn Féin fared best in terms of favourable responses; 39% of those polled felt the party's performance on this issue was good, compared with 26% for the DUP.
14% felt that the SDLP's performance was good, with 15% responding favourable to the Alliance Party. Only 10% felt that the Ulster Unionists had put in a good performance.
Following the personal revelations surrounding Peter Robinson and Gerry Adams, respondents were also asked whether the leaders should resign or remain in their current positions.
56% felt that the DUP leader should stand down and 37% said the Sinn Féin president should resign.
Over 1,000 adults in Northern Ireland took part in the survey.