Buckingham Palace said in a statement: "We are clearly very concerned about allegations in The Times following claims made by former staff of The Duke and Duchess of Sussex.
"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."
It is understood the Palace hopes to start the investigation soon.
Any changes in policies or procedures will be shared in the Palace's annual Sovereign Grant report.
The Times reported on Wednesday that a complaint was made against Meghan, 39, by a close aide of the couple in October 2018.
It came from the Sussexes’ former communications secretary Jason Knauf who suggested that Meghan made unreasonable demands that led to the departure of two personal assistants and the undermining of a third during her two-year period as a working member of the royal family.
The claims were not pursued and the following month Mr Knauf handed in his notice. Mr Knauf now heads the charitable foundation of the Duke and Duchess of Cambridge.
A spokesman for the couple has dismissed all the bullying claims out of hand and lawyers acting for them said they were based on misleading and harmful information dating back several years and deliberately leaked ahead of the interview.
A strongly worded and lengthy rebuttal to The Times from the couple’s legal advisers said: “Let’s just call this what it is — a calculated smear campaign based on misleading and harmful information.”
A spokesman added: “The duchess is saddened by this latest attack on her character, particularly as someone who has been the target of bullying herself and is deeply committed to supporting those who have experienced pain and trauma.
“She is determined to continue her work building compassion around the world and will keep striving to set an example for doing what is right and doing what is good.”
Meghan and Harry, 36, have frequently complained that she has been a victim of bullying, particularly at the hands of the British tabloid media