Spurs are scheduled to play Rennes in the Europa Conference League on Thursday and Uefa says “the match is due to take place as scheduled”.
They then play Brighton on Sunday in a Premier League game which is believed to be under review.
“Now, for sure, we are a bit scared,” Conte said.
“Every day, we are having more positives. This is not a good situation.”
Following the game against Brighton, Tottenham face Leicester City on 16 December and Liverpool on 19 December in the top-flight prior to an EFL Cup quarter-final against West Ham on 22 December.
“To speak about football today is impossible. The last situation made me very upset,” said the Italian.
“The situation is serious. There is a big infection.
“We prepare for the game against Rennes, but it is very difficult.
“Again at the end of the session, one player [tested] positive, another staff [member] positive – tomorrow, who [will it be]?
Conte added: “This for sure makes me upset because I am here to talk about football, about my players and the atmosphere that I want to see tomorrow.
“Instead we are talking about the people that have Covid.”
Conte, who said he had been vaccinated, did not say if the positive cases at the club were caused by the new Omicron Covid-19 variant.
On the state of vaccination among his squad, he said: “I think everyone takes [their own] best decision and is a question for the medical department about people as to whether they are vaccinated or not.”
Last season, Newcastle United’s game at Aston Villa in the Premier League was postponed following a “significant increase” in Covid-19 cases at the Magpies’ training ground.
Later that month, a coronavirus outbreak at Manchester City caused their Premier League match at Everton to be postponed four hours before kick-off.
Villa also had a Covid-19 outbreak in January, which meant that no first-team players were available for an FA Cup third-round defeat by Liverpool, prior to Premier League games against Everton and Tottenham being rescheduled