It follows consultation with his doctor, who is acting with "an abundance of caution", the source said.
Prince Philip, 99, remains in good spirits after being taken to King Edward VII Hospital in central London as a precaution on Tuesday.
The exact reason for his admission has not been disclosed.
However, his stay is not related to coronavirus.
A palace source previously said the duke, who turns 100 in June, had walked into the hospital in Marylebone unaided.
Last month, Buckingham Palace announced that Prince Philip and the Queen had received Covid vaccinations from a household doctor at Windsor Castle.
The couple have been spending the recent lockdown in England at Windsor with a small household staff, nicknamed HMS Bubble.
The Queen, 94, performed her first face-to-face event of the year on Thursday, when she knighted a royal aide during a private socially-distanced ceremony at her Berkshire residence.
The Duke of Cambridge carried out several virtual conversations with front-line NHS staff on Thursday, while the Princess Royal held a virtual meeting in her role as patron of the Scottish Rugby Union.
The Prince of Wales and the Duchess of Cornwall have also continued official duties.
In a pre-recorded video message for a virtual meeting hosted by the British Asian Trust, Prince Charles said on Thursday it was a "tragedy" that not everyone was benefitting from the "extraordinary achievement" of the coronavirus vaccine.
The Queen and Prince Philip celebrated Christmas quietly at Windsor, instead of the traditional family gathering at the Queen's Sandringham estate in Norfolk.
In December 2019, the duke was admitted to the same hospital - King Edward VII Hospital - as a "precautionary measure" for treatment for a "pre-existing condition".
He was discharged on Christmas Eve after four nights and driven to Sandringham, where he spent Christmas with the Queen.
He has received treatment for various other health conditions over the years including a blocked coronary artery in 2011, a bladder infection in 2012 and exploratory surgery on his abdomen in June 2013.
Prince Philip's most-recent official engagement saw him hand over his role of colonel-in-chief of The Rifles in a ceremony at Windsor to the Duchess of Cornwall, who received the role separately.
Last November, the Queen and Prince Philip marked their 73rd wedding anniversary by releasing a photograph of themselves opening a card made for them by their great-grandchildren, Prince George, Princess Charlotte and Prince Louis.