The first minister's stance on Brexit is "confusing" and "not sustainable", a senior Welsh Labour source has said.
On Wednesday, Mark Drakeford said "we're very near" the point a further referendum should be called.
But he stressed the priority should be to allow time for MPs to reach a consensus on a Brexit deal.
The source told BBC Wales Mr Drakeford should be "clearer in calling for another referendum". Mr Drakeford said the position was "perfectly clear".
The criticism of Mr Drakeford, on Friday, came as Prime Minister Theresa May tries to persuade MPs for a third time to back her Brexit deal.
The Commons will vote on her withdrawal agreement by 20 March, after MPs agreed to ask the EU to delay Brexit beyond the current 29 March departure date.
The source, who did not want to be named, said Mr Drakeford's lack of clarity at such a crucial time on the Brexit process was "causing some tension within the cabinet" of the Welsh Government.
"It's not sustainable to carry on saying that the view of the Welsh Government hasn't changed when so much has changed over the past two and a half years [since the referendum]," the source said.
"The position is confusing. He should be clearer in calling for another referendum."
However, Mr Drakeford said the position was "perfectly clear, and has been since we published Securing Wales' Future more than two years ago".
"We continue to believe that, in a genuinely free vote, there would be a majority in the House of Commons for the sort of Brexit advocated in Securing Wales' Future," the first minister said.
"In our view such an outcome would both respect the result of the 2016 referendum and minimise the economic damage that will be done by leaving the EU.
"We have long said that if Parliament concludes that a second referendum is the only way to break the logjam, we will support it.
"Given how little time now remains, we believe preparations to enable a second referendum should begin immediately, in case one is required."
During last year's Welsh Labour leadership campaign, the two candidates who challenged Mr Drakeford both supported another referendum on Brexit.
Vaughan Gething, the Health Minister and Eluned Morgan, the International Relations Minister, were strong supporters of a so-called People's Vote.
Former First Minister Carwyn Jones has also come out in support of a further referendum.