Toshiro Muto said he would keep an eye on infection numbers and hold “discussions” if necessary.
More than 70 people associated with the Games have tested positive ahead of Friday’s opening ceremony.
His comments came on the same day International Olympic Committee (IOC) president Thomas Bach said cancellation was “never an option”.
Earlier this month, Japan announced the Games would go ahead in empty stadiums – despite growing concern over the Covid situation.
“We will continue discussions if there is a spike in cases,” said Mr Muto when asked at a press conference if the Games might be cancelled even at this late stage.
“At this point, the coronavirus cases may rise or fall, so we will think about what we should do when the situation arises.”
Although the Games don’t officially begin until Friday, the sporting action has kicked off with the women’s softball and football competitions starting on Wednesday.
Earlier this week, two South African footballers staying in the athletes’ village tested positive for coronavirus. A Czech beach volleyball player later tested positive for the virus.
A total of 71 people accredited for the Tokyo Olympics, including those accredited for the Games, have now tested positive for the virus.
The 2020 Olympics and Paralympics were postponed for a year due to the coronavirus pandemic.
It’s the first delay in the 124-year modern history of the Games.
The opening ceremony of the Olympic Games is due to be held on 23 July, though the first event of the Games – a softball game between Japan and Australia – took place on Wednesday.
The Olympics will end on 8 August, with the Paralympic Games due to start on 24 August and to end on 5 September.
But there has been public anger over coronavirus restrictions.
People in Japan are also concerned about the influx of tens of thousands of people associated with the event and the impact they could have on infection rates.
Host city Tokyo is facing a fresh surge of infections, with 1,387 cases recorded on Tuesday.
Japan is currently under a state of emergency which will be in place until 22 August.
Organisers are “concentrating 100% on delivering successful Games”, a Tokyo 2020 spokesperson said after Mr Muto’s comments on Tuesday.