First Minister Nicola Sturgeon is warning that "all bets are off" if those involved in Scottish professional sport do not respect extra privileges.
A new stay at home message is now being enforced in Scotland amid rising cases of Covid-19.
But hundreds of untested football players and coaching staff will travel across the country this weekend for Scottish Cup second-round fixtures.
Ms Sturgeon said football bodies must listen to player concerns.
Several football clubs outside of the senior game have suspended training and the North Caledonian Football Association has put training and matches on hold until next month.
Asked at her daily briefing to respond to a call from Lowland League club Dalbeattie Star to suspend the season until the pandemic is brought under control, Ms Sturgeon said the Scottish government is continuing to discuss the issue. "We've got to listen" if players or staff do not feel safe, she added.
The first minister said rules allowing the continuation of professional sport, not just football, will be reviewed if current mitigating rules, including separate travel and social distancing, are not followed.
Ms Sturgeon earlier this week suggested that Celtic had questions to answer about their winter training break to Dubai, although the Premiership champions insisted it was approved by the Scottish government.
National clinical director Jason Leitch commented that safety concerns expressed by Dalbeattie and others must be risen "up the chain" and "that may end up" at the Scottish FA and SPFL.
He said the government is in constant communication with the footballing authorities about which clubs were included in the exemption of "professional football".
Prof Leitch said that if there is a "common feeling" about safety from a part of the league then it is "for them to take account of that".
Asked if it was acceptable for hundreds of untested football players to be travelling to Scottish Cup ties this weekend, he pointed out that some footballers "may have the virus and should behave as though they have".
Prof Leitch added that, while professional sport remains exempt from travel restrictions, clubs must have "strong mitigations" in place - for example, having as low a level of support staff as possible and keeping to social distancing and use of face coverings.
Dalbeattie Star, whose manager, Ritchie Maxwell, works in the NHS, said in a statement: "We are classed as elite professional players, but no provision is available for Covid testing at our level.
"We do, however, query the professional status as we are a part-time club and all players have other careers out with football.
"Like all clubs in the league, we would love to be able to be able to complete this season in a timely manner, but the safety of our committee, management team, players and of course the people of our local community is of more concern than football."
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