Wednesday's report comes days before the broadcast of an interview that Meghan and her husband Prince Harry, the Queen's grandson, have given to American chat show host Oprah Winfrey spelling out why they quit their official roles.
Some commentators have suggested they may also criticise their treatment by the royal family.
Harry (36) and Meghan (39) issued a statement denying that she had bullied anyone.
The Times cited unnamed sources as saying an aide to the couple had raised a complaint in October 2018, alleging that Meghan had reduced some of her assistants to tears and treated others so badly that they had quit.
The paper said Harry had urged the aide, who has now left their staff, to drop the complaint, and it never progressed.
The Times said it had been contacted by former staff members who wanted the public to gain insight before the Winfrey interview aired - and that lawyers for the couple had labelled the allegations a smear orchestrated by the Palace.
Reuters could not independently verify the report.
"We are clearly very concerned about allegations in The Times following claims made by former staff of The Duke and Duchess of Sussex," Buckingham Palace said in a statement on Wednesday.
The Royal Household "does not and will not tolerate bullying or harassment in the workplace".
It said its HR team would look into the allegations, and that members of staff involved at the time would be invited to participate.
'SADDENED BY ATTACK'
There was no immediate response from Harry and Meghan to the Palace statement.
A spokeswoman had earlier said Meghan was "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".
It added: "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, who married in May 2018, stepped back from their official duties in March last year to forge new careers and a financially independent life in California.
That decision was confirmed last month, when they also handed over all their royal patronages. They said their move was fuelled in part by intense press intrusion.
However, Meghan had also previously indicated that she felt she did not have the full support of the royal family.
Royal biographer Penny Junor told Reuters the Winfrey interview was likely to be frank.
"I suspect there will be snipes at the royal family. There will probably be a justification for their leaving, which the royal family will probably find uncomfortable," she said.
"My biggest fear, I suppose, is that Meghan and Harry say things about their family that they will regret."
DUCHESS WINS PRIVACY CLAIM
Meanwhile, Meghan was awarded £450,000 ($NZ856,000) as a provisional payment towards her legal costs after she won a privacy claim against the Mail on Sunday which had printed extracts of a letter she wrote to her father.
Last month, a judge at London's High Court ruled the tabloid had breached her privacy and infringed her copyright by publishing parts of the five-page letter she wrote to her father, Thomas Markle, who she fell out with on the eve of her wedding to Prince Harry.
Judge Mark Warby ruled in her favour without holding a trial, saying the articles were a clear breach of privacy after the paper argued the duchess had intended the letter's contents to become public and it formed part of a media strategy.
At Tuesday's hearing, Warby refused the paper permission to appeal that decision, saying he saw "no real prospect" that the Court of Appeal would reach a different conclusion. However, the paper is able to apply directly to the court.
The court was told Meghan's legal team was seeking more than £1.5 million in legal fees, with half the amount to be paid within 14 days, a sum the paper described as "disproportionate".
Warby agreed to make an interim costs order of £450,000, saying the final sum "may well be considerably more than that" after other outstanding issues were resolved at later hearings.
Meghan's legal team also demanded the paper hands over any copies it has of the letter, and called for the judge to order the paper to publish a statement on its front page stating she had won her case, with a notice also placed on the MailOnline's home page for "not less than 6 months" to act as a deterrent.
Warby said he would not make an order for the delivery or destruction of any copies of the letter yet.
Her lawyer Ian Mill told the hearing that they were not seeking to punish the paper, and would accept nominal damages based on the profits the Mail made from its articles, saying this was a "proportionate" way forward.
In its written submissions, the paper's lawyers wrote: "No purpose would be served by a hearing to determine the precise amount, which by definition is not relevant. It is suggested that 1 pound, 2 pounds or 5 pounds would do."
Meghan and Harry have regularly appeared on the front pages of Britain's newspapers in the last month, having announced they were expecting their second child, followed by news of their final split with the royal family.
