Royal sources have said Prince William is focused on looking after his wife, the Princess of Wales, who is recovering from abdominal surgery and is taking time away from public duties until after Easter.
Prince William is expected to carry out an investiture at Windsor Castle on Wednesday morning before attending a gala for the London Air Ambulance in the evening.
Prince Harry landed at Heathrow Airport at lunchtime on Tuesday on a flight from Los Angeles before travelling in a convoy of SUVs to central London.
He was pictured arriving at the gates of the royal residence at around 2.42pm.
Sources close to Harry, who now lives in California, have suggested he would have welcomed an opportunity to meet with William “if the opportunity were to arise”.
A source told the Mirror: “The Duke's primary reason to travel to the UK is to visit his father.
“If the opportunity were to arise to see the Prince of Wales then the Duke would have gladly accepted it.”
A source close to William also told the newspaper: “The prince’s main focus is for his wife, who is recovering from surgery, his three children and now his father.
“There are no plans for Harry’s visit to act as some kind of vehicle for reconciliation.”
Harry travelled without his wife Meghan, the Duchess of Sussex, who has stayed in the US with the couple’s two children, Prince Archie, four, and Princess Lilibet, two.
Charles and Harry spent time together at Clarence House before the King was pictured leaving the property with Queen Camilla in a car.
It was the first time the King has been seen in public since the announcement of his cancer diagnosis.
He and the Queen smiled and waved from his car to well-wishers as the pair headed to Buckingham Palace where it is understood they left in a helicopter to take them back to Sandringham.
They left around 45 minutes after Harry's arrival. Harry reportedly departed Clarence House for Buckingham Palace a few minutes earlier in a black 4x4. It is unclear if he has travelled to Sandringham too.
Charles, 75, is facing regular treatment after his shock diagnosis during his recent hospital stay for an enlarged prostate.