Among 704 members asked at the stadium before Sunday's game at home to Real Sociedad, or by telephone, whether the Portuguese's actions and statements were damaging, 61.6 percent answered yes, with 33.1 percent saying no, Marca said.
Members gave his performance as coach since he took over in 2010 an average mark of 6.68 out of 10, down from 8.82 in a similar poll conducted in March 2011, but 54.4 percent still thought he should stay on after this season, with 41.8 percent saying he should leave, they added.
The survey is the latest evidence that all is not well at Real, who are 16 points behind arch rivals Barcelona in La Liga and facing a possible exit from the King's Cup at home to Celta Vigo later on Wednesday.
It will also make painful reading for president Florentino Perez, who is up for re-election this year and has given Mourinho unprecedented powers in the hope he can lead the club to regular silverware in domestic and European competition.
Mourinho's combative style and regular insinuations of bias against his teams are seen as unsporting by many and at odds with what Real regards as its traditional image of a gentlemanly club that is gracious in victory and defeat.
He has come under fire recently for relegating hugely popular goalkeeper and captain Iker Casillas to the bench amid reports of dressing-room rifts between Spanish and Portuguese contingents in the squad.
Real need to overturn a 2-1 deficit against Celta in Wednesday's Cup last-16 second leg at the Bernabeu if they are to secure a quarter-final meeting with Valencia.