Some roads in Wales will revert to 30mph following backlash at the Welsh government's £34m default 20mph policy.
Transport Minister Ken Skates said the changes will address the concerns that "a lot of people" have raised "on a consistent basis".
"We've put our hands up to say the guidance has to be corrected," he said.
Swansea council leader Rob Stewart welcomed the change, but said the government must help foot the bill to swap signage.
Mr Skates highlighted the almost half a million signatures to a Senedd petition opposing the policy, brought in under former first minister Mark Drakeford and his transport minister Lee Waters.
"I have friends and family who have signed the petition," he said.
But the policy has also enjoyed support from groups including those representing cyclists, while some people have even called for it to be extended to areas not covered in the rollout to help make their communities safer.
Mr Skates was put in charge of transport for Wales on 21 March after Vaughan Gething succeeded Mr Drakeford.
"There is generally universal support for 20mph being targeted in areas where there are schools, built up areas like housing estates, and outside hospitals and so-forth," he told North Wales Live on Friday.
"But in many areas, routes that shouldn't have been included, were."
He said the changes will enable individual councils to revert routes back to 30mph where appropriate, adding that this will allow for "radical" changes, if that is "what people want".
Mr Skates is expected to address forthcoming changes in a statement to the Senedd on Tuesday afternoon.
Rob Stewart, leader of Swansea council, said Mr Skates' "pragmatic approach" was "to be welcomed".
However, he said the Welsh government should "help us with the cost" of reverting back to 30mph roads.
Mr Skates added that he does not feel it would be fair to place that costs on local authorities, given that councils are making "really tough" decisions.
"I'm not going to say to councils that they need to find the money to make the changes," he said.
The controversial 20mph policy was introduced in Wales in September with the aim of improving road safety. It saw the default speed limit on roads in built up areas cut from 30mph to 20mph.
Mr Stewart added that he anticipated the changes to the policy could affect up to 10 roads in his own local authority.
"I don't think anybody is challenging the requirements or the ambition of the 20mph to make our roads safer, but it's about making sure we've got the balance right," he said.
"So I think it's just about being pragmatic and doing it in a way that's sensible."
Huw Thomas, leader of Cardiff council, said the news was welcome and that it was not possible to "pretend there isn't a half-million-person petition".
He said he was glad the power will rest with local governments and that the work will be "mostly" funded by the Welsh government.
Mr Thomas added that in Cardiff, where most of city was 20mph already, the policy had been "very popular" and that he "certainly can't see many roads changing".
