Spurs are due to play at home to Rennes in their final Europa Conference League group game on Friday (AEDT), before travelling to Brighton and Hove Albion in the Premier League on Monday.
Speaking at a media conference ahead of the Rennes clash, an emotional Conte revealed the extent of the outbreak at the club.
"Eight players and five members of staff [have tested positive]," Conte confirmed. "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."
Conte had been scheduled to be joined by midfielder Pierre-Emile Hojbjerg at the news conference, but instead faced the media on his own.
"To speak about football today is impossible," the Italian added. "The situation is serious. There is a big infection.
"We prepare for the game against Rennes, but it is very difficult."
Tottenham currently sits second in Group G, and needs to match or better Vitesse's result against NS Mura on Friday to qualify, with Rennes having already won the group.