Two of the candidates in the Welsh Labour leadership contest say they are willing to reverse changes to funding for disabled people if there is evidence they are losing out.
Councils have been put in charge of support for 1,300 former recipients of the Independent Living Fund (ILF).
BBC Wales research found that around 100 people have had care packages cut.
Mark Drakeford said he would potentially reconsider the move - a call later echoed by Eluned Morgan.
Both are ministers in the Welsh Government, which had protected the money in the scheme until earlier this year when it scrapped the Welsh Independent Living Grant (WILG) and passed the responsibility to councils.
A source in Welsh Labour accused Mr Drakeford and Ms Morgan of breaching government collective responsibility, and said the matter would be raised with First Minister Carwyn Jones.
The ILF was provided by the UK government until 2015, when it was transferred to English councils and devolved governments.
Research by the BBC Wales Live programme showed about 100 of the 600 recipients who have been reassessed have had care packages cut.
Huw Irranca-Davies, the minister in charge, previously said he will not "rethink the policy in its entirety" and that the scheme had no "losers".
But at a leadership campaign event in Blackwood on Thursday, Mark Drakeford said if an independent evaluation "shows the new system is not working as well as the old one then I would be prepared to reverse it because this is money intended for a very specific number of people for a very specific purpose".
Mr Drakeford, who is finance secretary, blamed the UK government for "breaking up that part of the welfare state", saying the ILF had "more or less disappeared" in England.
"The money is the same as it always was, and most local authorities I believe are doing a decent job of continuing to hand the money on," Mr Drakeford said.
"But we are beginning to pick up information that, in some places, that is not happening and the money isn't going to ILF recipients in the way that it would have been last year."
Mr Irranca-Davies had agreed to the evaluation, Mr Drakeford said, adding that it would be carried out by someone "who is nothing at all to do with local authorities or the Welsh Government".
Eluned Morgan told BBC Wales she would also be willing to reverse changes to support for disabled people.
"As socialists we exist to protect the most vulnerable. It is part of our raison d'etre," the Welsh language minister said.
One campaigner, Nathan Lee Davies, said he suffered a "panic attack" when he saw Mr Irranca-Davies speak about the issue on Wednesday "as he once again denied the evidence of suffering that is staring him in the face".
His campaign group - 'Save WILG' - has had backing from Welsh Labour politicians and the Welsh arm of the left-wing Momentum organisation, Welsh Labour Grassroots.
Praising Mr Drakeford's comments, he said: "We welcome this move as a very positive development in our long-running campaign and it reinforces the view of most members of our campaign that Mark is the right person for the job of leader."
Welsh Labour's Spring conference passed a motion supporting a campaign to reinstate the Welsh Independent Living Grant.
Mr Drakeford and Ms Morgan's rival, health secretary Vaughan Gething, declined to respond.
The Welsh Government also declined to comment.