Royal aides refused to deny the incident, but The Sun said they both had put the incident behind them.
However, their relationship is now firmly in the spotlight with tabloid headlines in the UK and royal fans taking to social media to share their views and theories.
After Meghan's relationship with Harry was first revealed, it was widely reported she relied on Kate for advice and support as she joined the royal family. The pair thrilled royal watchers when they spent a day together at Wimbledon earlier this year.
A month after Meghan and Harry's wedding they appeared together at an event at Cheshire, in northwest England. They both attended Remembrance Day commemorations this month, but stood at separate balconies.
Last week it was revealed that the Duke and Duchess of Sussex will move out of Kensington Palace where they are neighbours of William's and Kate's and move 50km away to Windsor.
A friend close to the couple has claimed that the decision to move may be because a rift is developing between the two couples because "Kate and Meghan are very different people".
Speaking to the Mail on Sunday, the source added: "They don't really get on."
Kensington Palace confirmed that Harry and Meghan were moving out, but a spokeswoman explained that this was to facilitate preparations for having a baby.
They added that Windsor was a special place for The Duke and Duchess of Sussex, who also formed a part of the decision to make the move.
But a royal source told The Sun: "The initial plan was for Harry and Meghan to move out of their cottage in the grounds of Kensington Palace and into one of the main apartments.
"But there has been a bit of tension between the brothers."
The Queen warned Prince Harry over Meghan's behaviour and attitude before their wedding. Meghan initially wanted a tiara that featured emeralds, according to royal sources. But the future Duke and Duchess of Sussex were unhappy when told her first choice was impossible, as no one knew exactly where it came from.
A book by reporter Robert Jobson reveals Harry told staff before the wedding: "What Meghan wants, Meghan gets."
Meanwhile, palace insiders have pointed to tensions between Harry and William before May's lavish wedding.
An insider said that Harry didn't think William was making enough effort to welcome Meghan into the family, The Daily Mail reported.
According to the insider, the tension between the brothers eased when Prince Charles intervened and William and Kate eventually invited Harry and Meghan to spend Christmas with them.