Huw Lewis has set out new guidelines for universities who want to train new teachers in future as part of sweeping reform to the system.
In a speech in Cardiff, he announced a new two-year postgraduate course and greater subject specialism for primary school teachers.
Mr Lewis has already unveiled plans to extend education degrees to four years.
He eventually wants all teachers to be educated to masters level.
At the moment some universities work together in three teacher training centres in Wales.
But Mr Lewis said he had an open mind and universities outside Wales could end up training students.
New guidelines to "transform" the system follow an independent review in 2015 which found quality had "deteriorated".
Mr Lewis said it was his last major speaking event as minister "but the agenda does not stop" and was "vital".
"We will have a new teaching training system which is of such high quality and unquestioned rigour that it gives individuals a reason, in and of itself, for the very best people to get into teaching in the first place," he added.
Mr Lewis said teachers also wanted more professional support and challenge.
He said: "I have been clear that we must do more to accelerate improvement in initial teacher education provision across Wales.
"This is particularly important as we continue with our programme of radical education reform, focused on driving up standards across the board."
The 2015 review of teacher training coincided with a wide-ranging report into education in Wales.
Work to alter the curriculum is now under way after the government accepted the recommendations made by the Donaldson review, which will see computer programming and IT become as central to classroom teaching as literacy and numeracy.
The review recommended having six areas of learning and replacing the key stage system with a more seamless progression through school.
It is expected to be rolled out from 2021.
Geraint Davies of the NAS/UWT union gave a cautious welcome to the proposals and said he "fully appreciated the education system of the future will need student teachers of the highest possible calibre".
But he said future Welsh governments must ensure that the profession was an "attractive proposition for those students".
Conservative Shadow Education Minister Angela Burns said the proposals did not go far enough.
She wants a centre for education that will focus solely on initial teacher training and educational research.
Wales will have the "toughest" and the "best" teacher training system in the UK, the education minister has pledged.