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.

Boston beat domestic competition from Los Angeles, San Francisco and Washington DC to be put forward.
"Our goal is to host Olympic and Paralympic Games that are innovative, walkable and hospitable to all," said Boston mayor Marty Walsh.
The deadline for cities to apply to host the Games is 15 September.
The International Olympic Committee are due to make a decision in 2017.
The last time the USA hosted a summer games was in Atlanta in 1996, although Salt Lake City, Utah, hosted the Winter Olympics in 2002.
"I very much want to bring the games to the United States to share the incredible spirit of the games with another generation of Americans, and advance the Olympic and Paralympic movements," said American IOC executive board member Anita DeFrantz.
The US is the third country after Germany and Italy to officially launch a bidding process.
At an IOC meeting in Monaco in December, a 40-point action plan was voted on to revamp the sporting showpiece.
Potential hosts will find it easier and cheaper to bid under the "Olympic Agenda 2020" reforms and events can be held outside the host country or city.
The 28-sport cap for future summer Olympics has been removed - although there will be no changes to the Rio 2016 schedule.
Instead of a 28-sport limit, each summer Games will be restricted to 10,500 athletes and 310 events.
Wales will have the "toughest" and the "best" teacher training system in the UK, the education minister has pledged.
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Boston has been selected by the United States Olympic Committee as its candidate city to host the 2024 Olympic and Paralympic Games.