Royal household staff face pay freezes and projects have been halted amid the closures.
These include Buckingham Palace, which usually welcomes visitors during summers when the Queen is at Balmoral.
The Sun reported that Lord Chamberlain Earl Peel - the most senior official in the royal household - told staff that income is expected to fall by a third this year.
In an email seen by the newspaper, the Lord Chamberlain said the royal household was "not immune" to the impact of the pandemic and that coronavirus could leave a £17.8 million black hole in royal finances.
All but essential expenditure has been suspended, a recruitment freeze has been implemented and the annual pay review has been paused, he is thought to have told staff.
He reportedly said in an email: “We must therefore assume it could still be many weeks, if not months, before we are able to return to business as usual.
“There are undoubtedly very difficult times ahead and we realise many of you will be concerned.”
A Buckingham Palace spokesperson said: “The whole country is very likely to be impacted financially by coronavirus and the royal household is no exception.
“However, the time to address this issue will be when the full impact of all the implications of the current situation is clearer.
“At the moment the attention of the royal household is on ensuring it follows all the guidelines and supports the national effort in combating Covid-19.”
Last week Buckingham Palace confirmed it would remain shut to visitors for the rest of 2020, while those who have booked to explore the Queen's home will be refunded.
The 94-year-old monarch has been staying at Windsor castle with the Duke of Edinburgh and is expected to stay there for the foreseeable future.
Between July and September, while the queen is usually away in Balmoral, thousands of tourist from around the world enter the palace to see the opening of its state rooms.
It is believed to be the first time it will be staying shut during the summer months since it first opened its doors in 1993.
The Royal Collection Trust said it had taken the decision because of the difficulty ensuring the public’s safety during the coronavirus pandemic.
Frogmore House in Windsor and the Prince of Wales’s London base Clarence House will also remain closed in 2020.
Garden parties and investitures at the Palace have already been cancelled, Trooping the Colour will not take place in its normal form and the royals have swapped public appearances for video calls as they carry out their duties while working from home.
Coronavirus: Everything you need to know: