The players' unions in both Italy and Spain want the upcoming Champions and Europa League matches scheduled in their respective countries postponed amid the coronavirus outbreak.
The Champions League match between Atalanta and Valencia was played in Spain on Tuesday, with the Serie A side winning 4-3 to advance to the quarterfinals.
However, if the players' unions have their way the upcoming Champions League match between Barcelona and Napoli on March 18, and the Europa League matches featuring Inter Milan against Getafe and Roma versus Sevilla on Thursday would be postponed.
A joint statement from the players' unions in Italy (AIC) and Spain (AFE) to UEFA read: "The situation in Italy, Spain and throughout Europe regarding the serious health crisis due to the spread of the virus COVID-19 is forcing us to make difficult and, unfortunately, inevitable decisions.
"This week there are three fixtures scheduled between Italian and Spanish clubs.
"Valencia-Atalanta was played behind closed doors but today [Tuesday] the Spanish government has increased emergency measures banning all flights to and from Italy.
"Playing a game in Milan in the next few days will jeopardise the health of many people.
"Travel between Italy and Spain in the next few weeks will only get more complicated and difficult.
"There is a high risk that even next week's return games will have to be played under emergency conditions in both countries.
"AIC and AFE jointly call on UEFA that games between Italian and Spanish clubs be suspended until they can be played in safer conditions and once the emergency level has diminished in both countries.
"Coronavirus is becoming a Europe-wide issue and no longer merely an Italian one. We trust that a deep reflection on the full match calendar for European competitions be made in light of the news that gets worse hour after hour."
Also on Tuesday, Getafe president Angel Torres told Spanish radio station Onda Cero that his team would refuse to travel to Milan for Thursday's match against Inter and said they had asked for the game to be played in a different location.
He said: "We don't want to go to the focal point, get infected, and then have to suspend La Liga. We've asked for help from the RFEF to intervene. Tomorrow we'll train in the afternoon to give more time and we'll decide, but right now Getafe won't travel to Milan.
"If we have to lose the tie, well, it's an exciting thing but health comes first. I'm not going to take any risks. If we have to lose the tie, we'll do it with our heads held high."
Tuesday had already seen numerous cancellations and storylines emerge surrounding the spreading coronavirus in Europe, beginning with La Liga announcing that top-flight football in Spain would be played behind closed doors for at least the next two weeks.
The ban on supporters attending games also extended to the country's second division.
The Spanish Players' Association (AFE) has sent a letter to La Liga, the Royal Spanish Football Federation (RFEF) and the High Council of Sports requesting all games at all levels be suspended in the country.
Wolves had also petitioned UEFA to postpone their Europa League match away to Olympiakos saying: "The trip poses unnecessary risks to our players, staff, supporters and the families of all who travel, at such critical and uncertain times."
The petition came after Nottingham Forest and Olympiakos owner Evangelos Marinakis said he tested positive for the virus. However, that petition from Wolves was denied and the game will take place on Thursday.
On Monday, it was announced that Paris Saint-Germain's Champions League match against Borussia Dortmund on Wednesday would be played in an empty stadium.
PSG's star striker Kylian Mbappe had been held out of training this week with an illness and was tested for the coronavirus on Tuesday, but the results came back negative.
Juventus' game against Lyon next week also will be played behind closed doors. Chelsea's trip to Bayern Munich is also expected to be played without supporters, although there has been no official confirmation.
In France, the sports minister Roxana Maracineanu has said that until April 15, Ligue 1 and Ligue 2 matches (France's two fully professional football leagues) will either be played behind closed doors or with 1,000 spectators depending on the wishes of the organiser.
Germany's Bundesliga is also studying the possibility of playing all fixtures behind closed doors, with Borussia Monchengladbach vs. Cologne the first match to be officially announced with preventative measures in place.
And Dutch football has been cancelled this weekend in the southern province of Noord-Brabant, meaning: PSV Eindhoven vs. FC Emmen, Willem II vs. sc Heerenveen and RKC vs. FC Groningen are postponed.
Talks have taken place in England, too, about a contingency plan if the coronavirus continues to spread but there are no plans to place restrictions on supporters or cancel games in the short term.
On Tuesday, it was also announced that Republic of Ireland's Euro 2020 playoff against Slovakia would go ahead, but behind closed doors and later on in the day the France vs. Ukraine friendly game on March 27 and Germany vs. Italy and France vs. Finland friendlies on March 31 went the same way.
ESPN FC writer Gabriele Marcotti contributed to this report.