Three people in Cologne tested positive for the coronavirus, the German club confirmed, although they said the training "will continue as planned".
The Bundesliga is expected to become the first major European football league to return to competition.
Officials suggested a resumption on May 9, but the government delayed decision and a restart can now be May 16 or 23.
Those who tested positive are "symptom free" but will remain in quarantine for 14 days, the club said.
"The whole team and the technical team, together with the backstage team, tested Covid-19 on Thursday," added the statement.
"FC Koln will not confirm a name out of respect for the privacy of those affected. The training can continue as planned, in accordance with the hygiene and infection control measures that have been in place since April 6 in the group training.
"The prerequisite for this is that the relevant group of people is further tested, as stated in the DFL's Medical Concept of Sports Medicine / Task Force / Special Game Operation".
On Thursday, Chancellor Angela Merkel said that any decision on whether and when sports activities could resume would be adopted on Wednesday, May 6.
The league has been suspended since mid-March because of the global coronavirus pandemic.
"It is absolutely necessary that we be disciplined and follow health guidelines," said Merkel.
DFL has warned that many top flight teams will be in a "threatening to exist" financial position if the game is not resumed in June.
With nine games remaining this season, Cologne is in 10th place.