A decision is set to be made no later than Wednesday on whether Glasgow will move down to level two restrictions from Saturday.
The city is due to host a Euros fan zone with a capacity for 6,000 people from Friday week that will run for all 31 days of the tournament.
The area will be ticketed and subject to physical distancing rules.
The Scottish Government has previously confirmed that 12,000 supporters would be allowed to attend Euro 2020 matches at Hampden.
Scotland is due to host the Czech Republic at the ground on June 14 before a clash against Croatia at Hampden on June 22.
Concerns emerged yesterday suggesting the Tartan Army could yet be shut out of Hampden, with Scottish ministers reportedly “increasingly nervous” about large crowds gathering in the city amid high case numbers and the spread of the Indian variant.
But pressed on BBC Scotland's Sunday Show on when Glasgow could be removed from level three restrictions, Mr Swinney said: “That’s why we have to take the measures we’re taking to suppress the virus with door-to-door distribution of testing kits, the availability of vaccinations and testing opportunities.
“The fan zones are going to be very regulated environments with significant constraints about who can get in and what testing needs to be taken, combined with all the steps to suppress the virus ... I’m confident we can achieve that.
“We think this extra week was necessary to suppress the virus, but we’re optimistic to do that [lift restrictions] at the end of the week.”
Approval for the Glasgow fan zone is “subject to the state of the pandemic nearer the time”, but Mr Swinney said he believed it would go ahead.
The Scottish Licensed Trade Association has been among organisations scathing of approval for the Euros fan zone, saying it adds “insult to injury” while pubs, restaurants and bars have been limited to earlier closing times and being unable to serve alcohol indoors due to existing restrictions.
Glasgow is the only area in Scotland under level three of the Scottish Government’s five-tier coronavirus restrictions, meaning residents cannot meet others indoors at home, travel out of the city or drink alcohol indoors in pubs and restaurants.
First Minister Nicola Sturgeon said on Friday that Covid-19 case numbers in Glasgow remain “uncomfortably” high, driven by an outbreak of the Indian variant of the virus mainly concentrated in the city’s southside.
Mr Swinney’s comments came as a major milestone was reached in Scotland’s vaccination programme, with more than two million people now having received both vaccine doses.
Figures released on Sunday showed 3,234,311 people have received the first dose of a Covid-19 vaccination and 2,022,728 have received their second dose.
Scotland recorded 526 new coronavirus cases, but no further deaths of people with the virus in the 24 hours to yesterday, according to the latest figures.
Mr Swinney said the Scottish Government may ditch Covid infection rates as the determining factor in deciding whether further lockdowns are required.
The Cabinet secretary for Covid recovery stressed the government was still “coming to conclusions” about future restrictions in the wake of the vaccination programme and emerging new variants, but said the “dynamics” of dealing with the virus had changed.
He said the data was being scrutinised carefully, but given just six people were currently in intensive care in Scotland, the use of infection rates as a major factor could change.
"It [the infection rate] is a significant issue and reflects ongoing discussions we’re having about suppression of the virus,” he said.
“Given that such a large proportion of the population are now vaccinated and the earliest information we have about new variants is they do not appear to be presenting as acute health challenges as some of the previous presentations have – though that’s very early information we’re still looking at and coming to conclusions about – it does change the dynamics of decision making as to whether or not we have to always follow the level of infection cases as we have in the past, or because of the extent of the success of the vaccination programme and lower levels of hospitalisation, we can take a different set of decisions in the future.”
Mr Swinney added: “We’ve got to look very carefully at the data – and what’s happening in England with outbreaks, and there’s a lot of co-operation with Public Health England about the relationship between cases and hospitalisation and levels of seriousness of condition – but at this stage there’s very clearly a different dynamic at play.
"When you go back to January 4 when we took the decision to apply the much stronger level of restrictions very few people had been vaccinated, but now three million people have had the first dose, nearly two million have had the second, so that changes the dynamic.
“I'm hopeful that over time we may be able to move away from the use of lockdowns and severe restrictions on our freedoms to deal with increasing cases, and instead to take a more targeted approach, using our high-performing Test and Protect system, enhanced surveillance and local outbreak management to contain and control increasingly sporadic outbreaks.”
Mr Swinney said he understood the concerns of the hospitality industry as a result of lockdowns.
He said: “We will take steps to relax restrictions at the earliest opportunity, but we have to be mindful it doesn’t take much for the virus to run away from us again.
“We’ve seen it in Moray and in Glasgow and that causes more disruption. So if we can get in place an approach which suppresses the virus and keeps it suppressed we can take the decision to open up further sectors.”
Mr Swinney also revealed the first cross-party steering group on coronavirus recovery has already met and agreed its core priorities.
He said people would still need to “do their part” by wearing face coverings, physically distancing and getting tested even if they did not have symptoms.
High numbers of cases in parts of Glasgow showed Scotland was not yet free of the virus, he said, while the Indian variant proved that “we must continue to be on our guard”.
Speaking for the Scottish Conservatives, Murdo Fraser said Covid recovery should be the "unrelenting focus" for the next five years.
He said the impact of Covid-19 had been "devastating" in terms of lives lost and the wider impact on the NHS.
Mr Fraser said: "There is also a huge and largely hidden story about the social impacts of Covid-19 restrictions.
"A growth in loneliness and isolation, a huge negative impact on mental health, the effects of which we are not likely to fully understand for some years to come."
He said the government should have a strategy of suppression rather than elimination of the virus, saying: "Just as we've had to learn to live with seasonal flu outbreaks, we're going to have to learn to live with Covid in the future."
Scottish Labour leader Anas Sarwar stressed that coronavirus continues to be a "huge challenge" for the country, saying dealing with the pandemic must be "the defining mission of this Parliament".
But with Glasgow now Scotland's Covid hotspot and the only part of the country still under level three restrictions, Mr Sarwar warned there was a "real risk" of trust between the public and politicians breaking down.
He said: "I think there is a real risk that that level of communication and public trust and confidence is at risk of breaking in terms of what is happening in the city of Glasgow.
"We need to see urgent action that gives a proper route map, an exit plan, for the city of Glasgow."
He argued that rather than having "perpetual lockdowns", there should instead be clear criteria set out on when an area would face further restrictions, with "proper channels of communication" in place between the government and businesses in such areas.
Scottish Green health spokeswoman Gillian Mackay stressed the need to "build back stronger and better services" for both patients and NHS staff after the pandemic.
Liberal Democrat health spokesman Alex Cole-Hamilton called on ministers to start the public inquiry into the handling of pandemic "without delay".
He said: "That inquiry should not be about politics but about catharsis, about healing and learning from the remnants of this virus for any future pandemic that might visit upon our shores."