According to the Daily Mail the charity fund would cover the costs of renovating Number 11 along with Downing Street’s state rooms.
The proposal comes amid claims that Symonds carried out a costly makeover of the living space.
Where does the Prime Minister live?
Boris Johnson, wife Symonds and son Wilfred live in a four-bedroom flat above Number 11 Downing Street.
Johnson, along with predecessors Tony Blair, Gordon Brown, Theresa May and David Cameron, has opted for the Number 11 residence due to the greater size of the living space.
Technically, however, Number 10 Downing Street is classed as the prime minister’s official residence.
Traditionally Number 11 is the official residence of the Chancellor of Exchequer.
Current chancellor Rishi Sunak lives in a five-bedroom mews house in Kensington with his wife Akshata Murthy and their two children.
Why does Boris Johnson want to set up a charity?
Boris Johnson reportedly wants to emulate a scheme used by the White House.
The Washington charity is funded by private donors and allows for the upgrading of interior design, antiques and art. The aim of a Downing Street charity would be for the continued upgrading of Number 10 and Number 11 Downing Street on heritage grounds.
Insiders have told the Mail that the reason behind the prime minister favouring a charity fund is the cost of renovations to Number 11 Downing Street carried out by Carrie Symonds.
Johnson had reportedly explored asking Tory donors to fund the renovations.
Number 10 have not denied the attempt to set up a charity.
A spokesperson said: “The Downing Street complex is a working building, as well as containing two ministerial residences. As has been the case under successive administrations, refurbishments and maintenance are made periodically.
“Matters concerning works on the Downing Street estate, including the residences, are covered in the Cabinet Office annual report and accounts.”
What renovations has Carrie Symonds made to Number 11 Downing Street?
According to the Daily Mail Johnson has complained about the costs of renovations to Tory colleagues.
They claim that he has been quoted saying that renovations cost “tens and tens of thousands of pounds” and "over a hundred grand".
The Mail claim that Johnson is particularly concerned about the cost of wallpaper saying she appears to have ordered “new gold wall coverings”.
A friend of Symonds defended the renovations stating: "The makeover is appropriate for a building of such huge importance.
"Carrie has exquisite taste. It is classic, stunning, stylish and chic. She should be congratulated not criticised."