Barack Obama: Queen is among my favourite people
"She is truly one of my favourite people," President Obama said of the Queen at his Downing Street press conference.
The US leader's praise came on the day after the monarch's 90th birthday.
"I confess I've also come back to wish Her Majesty the Queen a happy 90th birthday," he said.
Ahead of a packed schedule, Mr Obama and his wife Michelle made a visit to the Queen at Windsor Castle their first stop during their three-day tour.
"Earlier today Michelle and I had the honour to join Her Majesty and His Royal Highness the Duke of Edinburgh as their guests at Windsor Castle where we conveyed the good wishes of the American people."
The Duke of Edinburgh, 94, drove the Queen into the castle grounds to meet the president and first lady as they disembarked from their helicopter.
After a quick chat, there was some discussion over where people should sit, with Mr Obama opting for the front seat.
Referring to Prince Philip, who drove the Queen and the couple from their helicopter to the castle, Mr Obama said: "I have to say I have never been driven by a Duke of Edinburgh before and I can report that it was very smooth riding."
This is the third time the couples have met and the first time in the historic surrounds of Windsor Castle.
Their last meeting was during a visit in 2011 when the Obamas stayed in Buckingham Palace.
In 2009, in what was taken to be a sign of a friendly relationship between the families, the first lady was photographed with her arm around the Queen.
In their meetings since, including today, there does seem to be a genuine warmth between the two families.
Mr Obama said the Queen had been "a source of inspiration for me like so many other people around the world".
"She is truly one of my favourite people and should we be fortunate enough to reach 90, may we be as vibrant as she is.
"She is an astonishing person and a real jewel to the world and not just to the United Kingdom."
There was no mention, however, of his birthday gift to the monarch, who received a photo album of her many meetings with US presidents and first ladies.
Another gift from Mr Obama to the Royal Family was a rocking horse, which was given to Prince George when he was born.
At the end of his first day in the UK, Mr Obama visited Kensington Palace for dinner with the Duke and Duchess of Cambridge and Prince Harry.
But before dinner, Prince George greeted Mr and Mrs Obama while dressed in his pyjamas and slippers.
Kensington Palace said the toddler thanked the president for the gift.