Coric beat Dimitrov in Croatia on Saturday before the latter returned to Monaco feeling unwell.
Zverev has since revealed negative tests across the board within his team, but the German apologised for his participation in the event.
The world number seven said on Twitter:
I have just received the news that my team and I have tested negative for COVID-19. I deeply apologise to anyone that I have potentially put at risk by playing this tour. I will proceed to follow the self-isolating guidelines advised by our doctors. As an added precaution, my team and I will continue with regular testing. I wish everyone who has tested positive a speedy recovery. Stay safe.
'We did all we could' - Adria Tour organisers react to Grigor Dimitrov's positive coronavirus test
Andrey Rublev, who was set to play Sunday's Zadar final against Novak Djokovic before it was cancelled, also said on Monday that he tested negative for COVID-19.
Marin Cilic also took to Twitter to say he will self-isolate and monitor his own situation after returning negative results.
Coric confirmed his positive test on Instagram, writing: "Hi everyone, I wanted to inform you all that I tested positive for COVID-19. I want to make sure anyone who has been in contact with me during the last few days gets tested! I am really sorry for any harm I might have caused! I'm feeling well and don't have any symptoms [sic]. Please stay safe and healthy! Lots of love to all!"
A statement from Adria Tour claimed they followed all safety measures after Sunday's cancellation.
The statement from Sunday read: "As a precaution and to protect the health of all participants and visitors, the organisers of Adria Team have decided to cancel the grand finale of the tennis spectacle planned for tonight.
"At all times we have strictly followed the epidemiological measures in place with the countries where the Adria Tour was organised. None of the people involved in the organisation and who have been in contact with Grigor have any symptoms. However, we are in close contact with health authorities to insure that all people who were in contact with Grigor Dimitrov will be tested tonight."
Dimitrov wrote on Instagram on Sunday: "I want to reach out and let my fans and friends know that I tested positive back in Monaco for Covid-19.
"I want to make sure anyone who has been in contact with me during these past days gets tested and takes the necessary precautions.
"I am so sorry for any harm I might have caused. I am back home now and recovering. Thanks for your support and please stay safe and healthy."