Prime Minister Mark Drakeford has said the best place to support players will be the couch, the local pub or the garden.
Otherwise, fans risk re-importing the coronavirus from countries where Wales play friendlies and group stage matches next month.
The team will warm up with a friendly match against France in Nice on Wednesday 2 June, before heading to Baku, the capital of Azerbaijan, for the group stages of Euro 2020.
There, they will face Switzerland (June 12) and Turkey (June 16).
The team then travel to Rome for the final group game on June 20.
France, Azerbaijan and Italy are all on the Orange List of countries for international travel to and from the UK.
The Foreign Office advises against all travel to Orange and Red List countries to reduce the risk of new Covid variants entering the UK.
Drakeford said Wales fans should now show off the same ‘great team spirit’ that pervaded the country during the pandemic.
He said: “Unfortunately, the pandemic is not over yet and that means we still need to make changes in our lives to stay safe.
“For football fans the best way to show your support for Wales is to stay in Wales and support the team from home.
“This is our very clear advice not to travel – France, Azerbaijan and Italy are on the orange list because the virus is circulating there.”
Football Association of Wales chairman Kieran O’Connor said the team had worked hard to qualify for the tournament and, after the long delay, anticipation would be high among fans.
He said: “We are all delighted that this tournament can finally take place.
“It’s also clear that playing in different locations during a global pandemic comes with certain challenges and responsibilities. “
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Mr O’Connor continued, “So we reiterate our call for the cooperation of the Red Wall not to travel to Baku, Rome or Nice in June, ensuring that our fans and our communities remain as safe as possible as we all play our part. role in navigating ourselves through this pandemic. “
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