Antonio Conte confirmed on Thursday (AEDT) that eight players and five members of staff have tested positive for COVID-19.
As a result of the spiralling number of cases, Tottenham has shut its first-team training centre.
UEFA rules state that teams must fulfil a fixture in the competition if they have 13 players, plus a goalkeeper, available for selection.
But Conte said Spurs only have 11 players to choose from and the Group G showdown with Rennes at Tottenham Hotspur Stadium has now been postponed.
A new date for the fixture, which Tottenham may have to win to remain in the competition, has not yet been set.
A statement on Tottenham's official website read: "We can confirm that our UEFA Europa Conference League Group G home fixture against Stade Rennais will not take place tomorrow after a number of positive COVID-19 cases at the club.
"Discussions are ongoing with UEFA and we shall provide a further update on this fixture in due course.
"Additionally, following discussions with the Health Security Agency (formerly Public Health England) and DCMS advisers, the club has been advised to close the first team area of its Training Centre at this time, in the interests of the health and safety of players and staff.
"All other areas of the Training Centre remain operational."
Tottenham's Premier League meeting with Brighton and Hove Albion on Sunday is also believed to be in doubt.
Speaking earlier on Wednesday at a pre-match news conference, Conte said: "Eight players and five members of staff [have tested positive]. But the problem is that every day we're having people with COVID, people that yesterday weren't positive and today are and we're continuing to have contact with. It's a serious problem.
"Today another player and another member of staff positive. Tomorrow, who? Me? Another member of staff? I want to speak about football. This is not a good thing for you, the fans and staff.
"Everyone is a bit scared, we all have families. I ask why? Why? Yesterday we trained and two positives, today we trained and after another two positives. Who next? It's not right, we're going home to our families.
"We had 11 players available for the match and by the end of the training session today one of the players who would start tomorrow is now positive. It's scary. We're all having contact."