Mr Martin visited the county at the weekend to address the party faithful.
But one senior member of the party told donegaldaily.com that the latest opinion poll “once again” raises the leadership issue.
“It’s a disaster. If there was an election tomorrow we would get wiped out and with council elections next we could lose a lot of seats in Donegal,” he said.
The Behaviour and Attitudes poll for the Sunday Times newspaper indicates that Sinn Féin have the backing of one quarter of the electorate, putting it firmly in second place.
The poll was carried out among just over 900 voters nationwide between Wednesday 15 February and Monday 20 February. It has a margin of error of plus or minus 3.3%.
Since the last Sunday Times poll in December, just after the Budget, support for Fine Gael has gone up by two points to 32%. Support for their coalition partners Labour has dipped by 1% to 10%.
Fianna Fáil’s support has dropped four points since December, leaving it at 16%. It is a result that will not please the party just a week before their Ard Fheis.
The big winner is Sinn Féin, who have gained at the expense of Fianna Fáil, up four points to 25% while the Independents were down one point to 17%.
Satisfaction with Taoiseach Enda Kenny is down three points to 41%, while Tánaiste Eamon Gilmore drops one to 34%.
Fianna Fáil’s Micheál Martin is down a substantial seven points to 33%, while Gerry Adams is down two to 46%, giving him the highest rating of any leader.
Satisfaction with the Government is unchanged, at 26%, while 70% are dissatisfied, and 3% have no opinion.