There have been almost 10,000 confirmed cases of the virus in the United Kingdom and close to 500 deaths.
League action has been put on hold until at least April 30, although the FA has extended the season indefinitely in a bid to complete the 2019-20 campaign.
And in a statement released on Thursday, English football's governing body outlined its commitment to its cup competitions.
Referring also to the non-league FA Trophy and FA Vase, the FA said of the FA Cup and Women's FA Cup: "We are reviewing all options as we seek to complete these competitions whenever it is safe and appropriate to do so.
"Clubs involved are close to reaching a major final and for those clubs and supporters we will do all we can to keep the Wembley dream alive."
The FA Youth Cup and smaller cup competitions will also be completed "if it is feasible to do so", while the FA is continuing with plans for next season's competitions.
While no decision has yet been made on how or when the Premier League and other elite competitions might conclude, the news came as the FA announced the end of several non-league seasons.
Results below the National League North and National League South have been expunged, with no promotion or relegation ahead of next season.
The same has been agreed for tiers three to seven of the women's game.