The first minister's comments came after Wales took a significant step towards reaching the last 16 of the competition with a 2-0 victory over Turkey on Wednesday.
Fan zones, where large groups of fans can gather to watch matches in designated areas outside, are usually a common site throughout the UK during football tournaments when the home nations are involved.
But they have been mostly axed this year due to the coronavirus pandemic, although fans have been gathering for matches at indoor venues.
Under current rules in Wales, up to 30 people can meet in any outdoor area. Regulated events of up to 10,000 people seated, or 4,000 standing, are allowed provided they pass a risk assessment, according to official guidance.
Speaking to BBC Radio Wales on Friday morning, Mr Drakeford said officials are exploring whether fan zones can be set up safely.
"We'll continue to be in conversations with local authorities who would be responsible for fan zones," he said.
"I think the argument is quite closely balanced, whether it is safer to have controlled environments where people can get together and enjoy the football in that way, or whether by having a fan zone you attract very large numbers of people and can't organise it in a way that keeps people safe."
Amid a rise in cases of the Delta variant, first detected in India, Mr Drakeford also announced a four-week pause to further lifting lockdown rules.
But the first minister said he will continue to consult public health experts on fan zones.
He added: "We continue to get the advice of the experts in that field, we talk to the police, we talk to environmental health officers, and we continue to discuss this matter with our colleagues in the local authorities as we hope Wales moves further into this competition."
Wales's next game in the Euros takes place on Sunday when the side takes on Italy at the Stadio Olimpico in Rome.