Buckingham Palace confirmed to The Sunday Times on Wednesday evening that the Queen sent a letter to Mr Biden, but did not reveal what was included in the message.
The Queen has met every US president, except for Lyndon B Johnson, since the beginning of her reign in 1952, with Mr Biden being the 14th American president to serve during her time as monarch.
Harry S Truman was president when the Queen succeeded to the throne on 6 February 1952. She also met a former president of the United States, Herbert Hoover, who served between 1929 and 1933.
When Covid-19 protocols allow, it is expected that Mr Biden will be invited to the UK for a state visit, as is the tradition between the two countries.
Despite serving as US vice president between 2009 and 2017 during former President Barack Obama’s administration, Mr Biden has not yet met the Queen, who has not travelled to the US since 2007.
When Mr Obama visited the UK for a state visit in 2009, he was accompanied by former first lady Michelle Obama, while the current US president stayed in the US.
Mr Obama and the Queen reportedly enjoyed a good relationship. He gifted her with an iPod featuring historic photos and videos from her visits to the US throughout her reign.
Donald Trump was the last president to meet the Queen on a state visit, in two trips that caused controversy and protests from UK residents.
Around 200,000 people protested against Mr Trump's state visit in 2018, as a 20ft “Trump Baby” balloon was raised in Parliament Square ahead of the trip.
When he arrived in the UK, Mr Trump then breached royal protocol by walking in front of the Queen, forcing her to step around him as he had blocked her way.
Mr Trump then breached royal protocol during a visit to the UK a year later when he appeared to lightly touch the Queen’s back. Royal protocol states that individuals should not touch the Queen, unless she has first offered her hand.
During the same trip, Mr Trump appeared to share an unusual handshake with the Queen, as he seemed to only grasp her fingers in his fist as he was welcomed to Buckingham Palace.
Protests once again took place ahead of Mr Trump’s visit in 2019, as then UK Labour leader Jeremy Corbyn boycotted a state banquet at Buckingham Palace, saying it would be wrong to “roll out the red carpet” for him, claiming that he had used “racist and misogynist rhetoric”.
Although Mr Biden has not yet met the Queen, Vanity Fair reported that he and US first lady Dr Jill Biden have met the Duke and Duchess of Sussex at several events.
Congratulations to @JoeBiden on being sworn in as President of the United States and to @KamalaHarris on her historic inauguration. America’s leadership is vital on the issues that matter to us all, from climate change to COVID, and I look forward to working with President Biden.
— Boris Johnson (@BorisJohnson) January 20, 2021
UK prime minister Boris Johnson also publicly congratulated Mr Biden after his inauguration ceremony on Wednesday.
“Congratulations to @JoeBiden on being sworn in as President of the United States and to @KamalaHarris on her historic inauguration,” Mr Johnson tweeted.
He added: “America's leadership is vital on the issues that matter to us all, from climate change to Covid, and I look forward to working with President Biden.”