Schools would need about three weeks to prepare for a phased return, according to First Minister Mark Drakeford.
In an interview on The Andrew Marr Show, Mr Drakeford mentioned June as an example of when some children could go back to school.
But this has caused confusion, according to a teaching union.
NAHT Cymru said: “At no point has June been mentioned in talks between Welsh Government and trade unions.”
The Welsh Government also clarified that no decision had been made on when schools would partially reopen.
Some schools have been closed for six weeks, although others have been open for the children of key workers and vulnerable pupils.
On Tuesday, Education Minister Kirsty Williams said there would be “a phased approach in allowing more pupils to return to school”.
Speaking on the Marr programme, the first minister gave more detail, saying: “We are thinking about ways in which we can bring young people with special educational needs back in to education.
“We’re thinking about particular year groups, Year 6 children in primary schools, children going up to secondary school this September.”
“We have a bilingual education system here in Wales.
“Children who are learning through the medium of Welsh and who may not have Welsh spoken at home, do we need to get those children back in to education sooner?
“Those are the sort of things we are working on at the moment,” he added.
On the issue of when schools in Wales might partially reopen, Mr Drakeford said: “Our advice from the trades unions and from the local education authorities is that we will need three weeks as a minimum from the point that we decide to do that to when schools can reopen, so we are talking about the beginning of June there.”
Speaking to BBC Wales after the programme, Mr Drakeford said he was using June as an example: “If we make the decision to return next week then the return to school will be in June.”
But he stressed that no decision had been made.
In an interview on the BBC’s Politics Wales programme, Counsel General Jeremy Miles said: “We’re not saying it’s the start of June.
“We’re saying there needs to be a lead in time so that schools and local education authorities can adjust to that.”
Teaching union NAHT Cymru tweeted that “at no point has June been mentioned in talks” between Welsh Government and unions.
“Speculation on dates is unhelpful,” it said.
It went on to say that the “profession must be at the heart of reopening decisions – the health and wellbeing of staff and pupils is paramount”.
Eithne Hughes, director of the Association of School and College Leaders Cymru, said: “We would caution against fixing a date in stone at this stage, and to make sure the conditions are right first of all.
“We are happy to plan towards a proposed date, but we would urge that it is clearly stated from the outset that it is moveable if more time is needed.
“We welcome the thought and care that the Welsh Government is putting into the reopening of schools.
“And we agree that the only realistic approach to doing this while sustaining social distancing is through a phased approach, in which certain groups of children are brought in first,” she added.