The competition had been due to start this weekend but with 10 positive tests at Kobe Steelers on Thursday taking the total to 67 league-wide, organisers said they were now aiming for a mid-February start.
Last year's season was cancelled entirely because of the pandemic.
"Japan Rugby Football Union and Top League have had discussions and we want to do everything we can to have this year's Top League season," said JRFU Chairman Kensuke Iwabuchi.
Organisers said they were yet to finalise a new start date and whether the format needed to be changed because of time restrictions and safety precautions.
Top League chief Osamu Ota hoped the season would finish by the scheduled end date of May 23 so that it would not interfere with the national team's plans.
"We are aiming to start in mid-February," said Ota.
"Regarding the COVID-19 situation, it changes on a day-to-day basis."
Kobe are the sixth team to report positive COVID-19 results along with NEC Green Rockets, Toyota Verblitz, Suntory Sungoliath, Canon Eagles and Toshiba Brave Lupus.
They said in a statement on their website that all team members, including those who had tested negative, had been told to stay home since Tuesday, when the positive test results were confirmed.
Matches involving Toyota, Suntory and Canon scheduled for this weekend had already been cancelled but the entire slate of first round games have now been wiped.
The JRFU said last week games would be played in front of thousands of fans who had already purchased tickets, despite government guidelines limiting the number of people allowed to attend sporting events.
Organisers suggested that policy would now change but did not provide details.
A host of top international players, including Beauden Barrett, Michael Hooper and Kieran Read, are set to play in this season's Top League, the final campaign before it is overhauled next year.