In response to a question from MSP Stuart McMillan at the Scottish Parliament’s Covid-19 committee on Wednesday, Ms Sturgeon said she would make a decision as to whether the championship’s games, which are due to take place in the summer, would go ahead.
The Scotland national team qualified for the rescheduled European Championships in November.
Hampden Park is set to host four games – three group stage fixtures, including Scotland's matches with Czech Republic and Croatia, as well as a final 16 clash.
Ms Sturgeon said: “Discussions are underway, not just for Scotland, but all of the countries hosting matches are grappling with the same thing.
"I really, really want the matches to go ahead at Hampden and I would love for them to go ahead with some spectators there to see them. We will do everything to bring that about.
"Obviously sitting here right now in March, it’s not possible to give hard and fast definitive answers to this, but there is no suggestion other than that we want these matches to go ahead and I would hope to think I’ll be there, cheering on Scotland, at Hampden, with more than a few others alongside me – if that is safe and possible to achieve.”
Tournament host cities have been told to give their most realistic assessment of stadium capacities by April 7.
Ms Sturgeon also gave renewed hope for football fans when she said she hoped spectators could return to football “soon” – but refused to specify a date.
In response to a question from MSP Willie Coffey, Ms Sturgeon said: “I hope that before too long, in some competition or another, fans will be able to be in Somerset Park to watch Ayr United beat Killie.
"Obviously what happened at the weekend puts huge questions in people’s minds about whether that was the right thing to do or not.
"We want to get sport back to normal, as we want to get everything back to normal, but one of the things Covid loves most is crowds of people, so we need to do it carefully and cautiously.
"But if we keep the figures going in the right direction just now, we can be really hopeful that all of these things are not too far in the future.”
Ms Sturgeon said she believed Police Scotland had taken a “sensible, proportionate and successful” approach to enforcing regulations, despite the scenes in Glasgow at the weekend after Rangers' Scottish Premiership win.
She said: “The vast majority of people have complied with all the restrictions.
"The minority who haven’t, probably the majority of them have breached out of error or ignorance of the regulations. There will be criticism of the police at times as there was at the weekend, but I think they have taken a sensible, proportionate and really effective approach to all of this."