Barnier has also gone into self-isolation, The Guardian reported, and physical meetings in the Belgian capital have been suspended, though online discussions continue.
The exact time when the negotiations will be resumed is uncertain. Under British regulations, the quarantine period for someone who has been in contact with someone who has tested positive for COVID-19 is 14 days, which is four days longer than the period under Belgian law.
Frost also tweeted about the negotiations.
"I am in close contact with @michelbarnier about the situation," Frost tweeted. "The health of our teams comes first. I would like to thank the @EU_Commission for their immediate help and support."
The two sides are trying to reach an agreement by next week to allow time for the European parliament to ratify the deal before Britain leaves the European Union's single market and customs union at the end of the year.
Agreement is still pending on post-Brexit trade issues like fishing, leveling the playing field for businesses and the governance of the trade agreement.
The talks were also delayed in March when Barnier and Frost both contracted the coronavirus.