The Sulphurites were due to host Rochdale in the competition this evening, before League Two trips to Crawley Town and Leyton Orient.
They have not revealed how many players have tested positive or how many are self-isolating after coming into contact with them, but the league routinely investigates the circumstances behind any postponements caused by the virus.
Last season Harrogate were forced to postpone matches because of the weather, and even had to abandon one due to a power outage, further congesting an already busy fixture list.
In the same campaign Orient were forced to forfeit their third-round tie against Tottenham Hotspur after a coronavirus outbreak left them unable to play the tie. Last term the first four rounds were condensed into September, and although this season's schedule is more forgiving, it is still tight.
The second round is due to be played on the week beginning August 23, and Harrogate have already cancelled their trip to Brisbane Road the previous midweek, leaving little scope to play the first-round game.
The league has tried to be as flexible as possible with regard to Covid-19 call-offs, and will consider whatever proposal Harrogate put forward before asking them to forfeit.
Harrogate actually beat the same opponents at the same venue on Saturday, starting their League Two season with a 3-2 win.