Ms Sturgeon would not go into details but she gave fans hope they could be in Glasgow for games against Czech Republic on June 14 and Croatia eight days later.
At her daily briefing, the First Minister said: "We have to put our proposals into UEFA, they have to then assess those and it will be for them to set out the proposals in due course.
"We are well aware of the deadline and while these are not final decisions for us, I am optimistic we will have matches played here in Scotland with a number of spectators, a reasonably good number of spectators in Hampden for them.
"But obviously these are discussions and deliberations and decisions that have to be made by the appropriate authorities."
UEFA has asked for the 12 host cities - including Glasgow, Dublin and London - for their proposals on how many fans they expect to be able to accommodate by Wednesday.
While the UK Government has announced plans for a number of test events - including a 21,000 crowd for the FA Cup final on May 15 - with a view to having Wembley full for at least some European Championship games, the Scottish Government has been more restrained.
There is no date set for fans to return in Scotland but provisional proposals will be handed to UEFA this week.
The SNP leader added: "I have said this before and I am not alone here, there are many people who will feel it even more strongly than I do, I desperately want to have these matches played here in Scotland at Hampden, not least because we will have the ability to cheer on Scotland in a couple of those matches.
"So it is really important for sports fans but also, just the idea that that will be possible come June, I think gives us all a lift and a belief that things might be getting back to normal.
"In terms of the European Championship, there are other countries involved in this, and some of those countries have real challenges on their hands with Covid right now. So there are some tough issues for all of us to continue to grapple with.
"I can't stand here in early April and give absolute 100 per cent guarantees for June on anything. That's not the nature of what we are dealing with.
"But I'm very, very hopeful I might be at Hampden to watch Scotland in the European Championship and we'll continue to work as hard as we can to make that happen."
Scotland have arranged two warm-up games ahead of their opener against the Czechs.
Steve Clarke's side will face Holland in the Estadio Algarve, near Faro, Portugal, on June 2 and Luxembourg away on June 6.
A Scottish Football Association statement added: "Further details of Scotland's preparations will be communicated in due course and in consultation with the Scottish Government with regard to elite athletes' training camp guidelines."
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