The Teacher Training Agency said there is enough flexibility to allow training providers to do both. But lecturers say that the regulations give students the impression that only the core subjects - English, maths and science - are important.
They argue that this decreases the demand for other subjects from students, making it less likely that institutions will offer them.
Roy Hughes, chairman of the Historical Association's primary committee, said: "The whole structure of the profession is geared towards developing the core subjects. The so-called commitment to a rich curriculum is vanishing."
At present trainee primary teachers have to specialise in one subject, but from September this will no longer be necessary.
The University of the West of England dropped the history specialism from its programme of undergraduate teacher-training degrees two years ago. By then the course was attracting fewer than 10 students.
Dr Penelope Harnett, principal lecturer in primary education at the University of the West of England, said: "We will offer history and geography - it is not good for children to learn from a teacher with no experience. But it seems to be a fact of life that humanities do not have the status of core subjects and the new standards are reinforcing this. Young teachers who want promotion will probably focus on core subjects."
A Geographical Association survey found geography has been dropped as a subject specialism by more than one quarter of initial teacher-training institutions.
Dr Margaret Mackintosh, former chair of the GA's primary and middle school committee, said: "I certainly feel we don't need so many hours devoted to literacy, numeracy and science in teacher-training courses."
Professor Colin Richards, chairman of the National Primary Teacher Education Conference, said: "Under the new standards, students may not have any history or geography at all. It is a very serious situation. Students in teacher-training college may not have done history or geography beyond 14 and that makes it even more important that they have some on their college course."