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Niall Carson/PA Wire
The Welsh Government has written to Ryanair asking the airline to postpone flights scheduled for Friday.
The budget airline is advertising flights to Malaga and Faro from Cardiff Airport.
A Welsh Government spokesperson said: "We don't believe these flights should be going ahead."
Ryanair has been asked to comment and Cardiff Airport said its airlines would "slowly re-start flying passenger services, increasing in August".
People in Wales have been asked to stay local, within five miles, as guidance.
These restrictions are expected to be lifted from Monday 6 July so people can "travel as far as they like for all purposes."
Tourism in Wales is not due to re-open until the following week on 13 July.
From 6 July people in England can travel to some European countries without having to spend 14 days in quarantine on their return but no decision has been made on this in Wales.
The Welsh Government said to slow the spread of coronavirus, the law in Wales requires people to "stay local and not to travel further than five miles unless they have a reasonable excuse".
It said: "While there are legitimate reasons why flights can operate, the onus is on individuals to obey the rules."
It said all travellers entering Wales from overseas need to self-isolate for 14 days to prevent the further spread of the virus.
Cardiff Airport said it had remained open throughout the pandemic to support essential flying including critical cargo and medical flights.
It added: "During July our airlines will slowly re-start flying passenger services, increasing in August.
"All airlines are making decisions about reinstating flights on a global basis and the situation remains fluid.
"We continue to follow government guidance and work closely with Public Health Wales to keep the safety of our team and customers as our number one priority."
Cardiff Airport was sold to the Welsh Government for £52m in 2013.