Health Minister Robin Swann has said he can see no reason why Northern Ireland's cemeteries could not be reopened on a controlled basis.
Cemeteries are operated by councils and have been closed, except for burials, since the lockdown began last month.
The Assembly has the power to reopen them via legislation.
Many bereaved families have called for the decision to be reversed to allow them to visit the graves of their loved ones.
Parties are split on the issue, with the DUP and Ulster Unionists in favour of reopening and Sinn Fein and Alliance opposed.
Mr Swann was asked about the issue at Wednesday's daily Covid-19 briefing at Stormont where he urged his Executive colleagues to allow the reopening of cemeteries.
The UUP minister said he saw no reason why people should be prevented from visiting graveyards if there was a system in place to ensure access was managed in compliance with social distancing rules.
He said: "In regards to cemeteries, as Health Minister I would see no reason why people couldn't be visiting cemeteries at this minute in time.
"There is a solace, a support there that many individuals can gain from visiting a cemetery and it's something when we reflect on where we have been and where we are now. It's a regulation that I think we as an Executive should actually challenge and change.
"Because we have to realise the mental challenge that we're putting on many individuals by asking them to remain in lockdown, and some will receive solace and support by visiting the graves of loved ones. It's something that I hope the Executive can come to a reasonable position on where we can open up cemeteries in a managed way.
"Should it be in a one-direction approach round the cemeteries in a single file or the number of people entering cemeteries or the actual time of day that cemeteries are open.
"I think it's an approach that we as an executive should be looking to," the North Antrim MLA added.
Northern Ireland's Chief Medical Officer Dr Michael McBride, who also attended the briefing, said that while his role was to provide advice to the Executive, any such decisions were up to Stormont ministers.
Meanwhile, two leading clergymen have advocated a cautious approach to be taken on the reopening of cemeteries.
Rev Steve Stockman from Fitzroy Presbyterian Church in south Belfast said health and safety has to be the main priority.
"In an ideal world it would seem almost pastorally sensible that people could go and visit their loved ones' graves," he said.
"The overriding factor has to be the health of the nation, however, and we may have to make sacrifices for the good of all because sadly cemeteries can be used in certain parts of the city for recreation and the gathering of young people."
But Rev Stockman also suggested that the introduction of supermarket-type crowd control measures could be a possibility.
"If it was managed and manned like going to the stores in that we let a certain number of people into the cemetery at a time then perhaps we could consider it," the minister said.
Fr Gary Donegan said he was against any immediate reopening of cemeteries and said it should only be considered when social distancing measures are relaxed and schools and workplaces reopen.
"Cemeteries should be kept closed for the time being," the priest said.
"When life starts heading back towards normality, that's when they should reopen.
"As difficult as it is not to be present at a grave, think about the doctors and nurses dealing with the victims of this virus.
"Think about their faces, where the elastic has cut in, with probably inadequate protection anyway.
"We have done very well with social distancing so far and that's why the death toll is so much lower than predicted. To prevent a second phase of this coming in we all ultimately have to make massive sacrifices."
Two church leaders have also called on the Executive to consider reopening cemeteries here in a controlled manner.
The Church of Ireland Bishop of Derry and Raphoe, Rev Andrew Forster, said that decision should "be looked at again".
The Catholic Bishop of Derry, Dr Donal McKeown, agreed, saying "it is a very difficult area", but asked the Executive to give "some sense of logic" for their motivation behind this particular piece of legislation.