A survey involving 1,000 members of the Association of School and College Leaders (ASCL) found that 24 per cent were opposed to the idea, while 67 per cent felt strongly opposed.
The reform was announced by education secretary Nicky Morgan in June and is to apply to pupils starting secondary school this month. It means they will have to study English, maths, science, history or geography, and a language up to GCSE level.
Concerns raised by respondents include not having enough room for creative and vocational subjects and that the EBacc would not "suit the needs" of every pupil.
Three-quarters of the school leaders, however, said they would be more likely to back the idea if there was more flexibility in the choice of subjects.
Elsewhere, the survey also found a shortage in the numbers of teachers within the EBacc subjects, which might also cause schools problems.
The subjects causing difficulty included languages, where 69 per cent of respondents said they faced recruitment difficulties, maths (59 per cent), and science (58 per cent).
The Department for Education has promised to consult with the profession this term over the exact detail and implementation of the reforms.
ASCL's deputy general secretary Malcolm Trobe welcomed this and is hopeful the consultation might lead to more flexibility in the system.
He said: "It needs to be recognised that the EBacc will not suit some pupils whose interests and talents may lay in other areas, and who will be demotivated by being forced to take GCSEs in which they have little interest. We hope that ministers will not therefore require that every pupil takes the EBacc and will allow that a proportion are better served by other options.
"We also think that the range and choice of EBacc subjects is too restrictive. It should be widened, particularly to include other humanities such as religious studies, and there should be greater freedom over the combinations of subjects that pupils are allowed to take."
Elsewhere, ASCL members were also asked how manageable they had found the current examination reforms, including the phased introduction of new GCSEs and A levels, the "decoupling" of A and AS levels and the new grading system for GCSEs. Around 55 per cent said the changes had been "difficult to manage but we have coped", 33 per cent said the changes had been "unmanageable and there is still much work to do", and four per cent said the changes were "unmanageable and we will not be ready in time".