Footage of the pair's meeting showed Mr Trump lose out in a battle of the handshakes when the Countess pulled the President towards her.
In the video, Mr Trump appears to try and pull away however the lady-in-waiting does not let go of his hand.
The president and First Lady Melania Trump met the Queen for tea at Windsor Castle as tens of thousands of people protested his visit to the UK on the streets of London.
The couple were treated to a Guard of Honour as they arrived at the royal residence on Friday afternoon after keeping the monarch waiting for several minutes.
The Queen was joined by Lieutenant Colonel Sir Andrew Ford, Comptroller, Lord Chamberlain's Office, and the waiting Guard of Honour - the 1st Battalion Coldstream Guards - gave a Royal Salute and the national anthem was played.
Mr and Mrs Trump were then invited to inspect the guard of honour in their red suits and bear skins. At one point, he stepped out in front of the Queen and then immediately stopped, forcing her to walk around him.
The couple were due to spend half an hour with the Queen but stayed longer than expected after their afternoon tea overran by 20 minutes.
In an interview with Piers Morgan, the president described the Queen as a “beautiful” and "fantastic" woman following their meeting.
The president told the controversial presenter: "The Queen is fantastic! She's a fantastic woman; so much energy and smart and sharp.
“She was amazing! Such a wonderful lady and so beautiful! It was such an honour to finally meet her. To have a Queen like that is great."