The allegations were made against the Duchess of Sussex in an article by The Times yesterday, in which the complaint claimed Meghan "drove two personal assistants out of the household and was undermining the confidence of a third staff member".
The complaint was reportedly made in October 2018 by Jason Knauf, who was Harry and Meghan's communications secretary at the time.
Meghan has denied allegations of bullying after a complaint was made by one of her advisers during her time at Kensington Palace, saying she is "saddened by this attack on her character".
Earlier today, less than 24 hours after the bullying allegations went public, Buckingham Palace released a statement saying it will investigate the claims.
"We are clearly very concerned about allegations in The Times following claims made by former staff of The Duke and Duchess of Sussex," the statement said.
"Accordingly, our HR team will look into the circumstances outlined in the article. Members of staff involved at the time, including those who have left the Household, will be invited to participate to see if lessons can be learned.
"The Royal Household has had a Dignity at Work policy in place for a number of years and does not and will not tolerate bullying or harassment in the workplace."
The Palace's decision to go public with its investigation has caused outrage among royal followers, with Prince Andrew's name quickly trending on Twitter.
"Let me get this straight. So, Buckingham Palace is investigating a bullying allegation against Meghan, but not the alleged sex abuse allegation by Prince Andrew victims. Did I get that right?" one twitter user wrote.
"Buckingham Palace (and royal family), where can we find the results of the investigation you did into this by Prince Andrew? We are waiting," wrote another.
"So, the Queen's team is conducting an investigation into the claims made against Meghan Markle. Where was the Queen's investigation into Prince Andrew or was she more than happy to sweep that under the carpet?" another wrote.
"With the Palace putting out such a strong statement about talking bullying regarding Megan (sic) so seriously. I'm assuming Prince Andrew is now booked on the first flight to the USA to clear things up with the FBI," yet another Twitter user wrote.
In 2019, the Duke of York announced he would step back from public duties after facing growing backlash over his relationship with disgraced financier Jeffery Epstein.
Despite the scandal, the Queen is yet to strip Prince Andrew of his military honours, as has happened to Prince Harry since his resignation as a senior royal.
Epstein was charged in 2019 with sex trafficking and found dead in his jail cell in early August that same year.
Prince Andrew was forced to give up his public royal role after a "car crash" BBC interview about his involvement with Epstein and allegations the royal himself had sex with an underage girl.
Andrew claims to have first met Epstein in 1999 through mutual friend Ghislaine Maxwell, then the financier's girlfriend and Andrew's old uni friend.
From the outset, the prince has denied any knowledge of Epstein's sex trafficking. He has also denied Giuffre's claims that he had sex with her while she was underage.