Marinakis, 52, was at the City Ground for Forest’s Championship fixture against Millwall on Friday.
Olympiakos are due to host Wolves in the Europa League on Thursday, although the BBC understands Wolves have asked for the game to be postponed.
“Mr Marinakis was diagnosed after showing the first symptoms on his return to Greece,” said Forest.
“During his stay in Nottingham, he did not show any symptoms of the virus.
“The club are seeking advice from medical professionals and the relevant governing bodies to ensure the correct measures are taken.”
Marinakis had earlier revealed he had been diagnosed with coronavirus on social media platform Instagram.
“The recent virus has ‘visited’ me and I felt obliged to let the public know,” he said. “I feel good as I take all the necessary measures and I discipline to the doctor’s instructions.”
Millwall, who won Friday’s game in Nottingham 3-0, have said that “senior club representatives who came into contact with Mr Marinakis have begun a period of self isolation”.
A club spokesman added: “This is purely a precautionary measure and no individual has displayed any symptoms. The club will continue to follow all necessary guidance from the appropriate bodies.”
The Lions are due to host Derby County at The Den on Saturday.
It was confirmed on Monday that Wolves’ last-16 first-leg match at Georgios Karaiskakis Stadium was to be played behind closed doors because of the coronavirus outbreak.
The Premier League club had sold 1,000 tickets for the fixture, with full ticket refunds currently being processed.
Wolves manager Nuno Espirito Santo has said: “If we have to go we will. But we don’t agree – we’re not happy to go.”
The Greek government announced on Sunday that all professional sports events for the next two weeks will be played without spectators, although Culture Secretary Oliver Dowden has said that events in Britain are unlikely to be affected by coronavirus in the immediate future.
Forest, who are fifth in the Championship table and scheduled to play Sheffield Wednesday in their next league game on Saturday, have been in contact with the English Football League and discussions are ongoing.
‘Nottingham Forest keen not to cause panic’
Forest are waiting for advice from health officials before taking any further action.
They say Marinakis was showing no symptoms of the virus while he was in Nottingham, which may reassure members of the public who might have come into contact with him on Friday.
It is not yet clear how many staff members or players he came into direct contact with but, as you would expect, the club is being bombarded with questions about how this might affect Saturday’s game against Sheffield Wednesday or future matches.
The genuine answer at the moment is that nobody knows, because they are awaiting medical advice. Once that advice is received, I would expect Forest to follow it to the letter, whatever the consequences regarding football.
At this stage, though, any speculation would be just that, and Forest are keen that they are not seen to cause panic or to be scare-mongering.
Marinakis was also in attendance at Emirates Stadium in late February when Olympiakos beat Arsenal in the previous round of the Europa League.
An Arsenal statement read: “Our home match with Olympiakos in the Europa League was played on Thursday, 27 February and none of our staff who came into contact with Mr Marinakis on that matchday have reported any symptoms since.
“We continue to follow strict protocols with regard to coronavirus. We’re following the government health guidelines and have additional procedures in place to protect our players and staff. This has included regular deep cleaning of areas used by players at Emirates Stadium and our training centre.”