their relationship was damaged by bullying.
A statement on behalf of Princes Harry and William described the story's "inflammatory language" as "offensive".
The Times quoted a source saying the Sussexes felt they had been pushed out and "constantly bullied" by William.
The brothers are to meet the Queen and Prince of Wales later to discuss Harry and Meghan's future.
The couple want to step back as senior members of the Royal Family, while also becoming financially independent.
A statement on behalf of Harry and William said: "Despite clear denials, a false story ran in a UK newspaper today speculating about the relationship between the Duke of Sussex and the Duke of Cambridge.
"For brothers who care so deeply about the issues surrounding mental health, the use of inflammatory language in this way is offensive and potentially harmful."
Claims of bullying have been strongly contested by other sources close to the Cambridges and Harry, the Times article reported.
In October, Harry responded to speculation about a rift between the brothers by saying the pair were "on different paths at the moment" but he loved William "dearly".
Dan Wootton, the executive editor of the Sun who broke the news of Prince Harry and Meghan's decision to step back, said: "It is quite extraordinary that at a time when Prince William and Prince Harry are having so much warfare behind the scenes that they have come together to issue this joint statement."
He told the BBC that while he would not use the word "bullying", Prince Harry and Prince William's relationship has become "incredibly strained over the past 18 months", and that has played "a big part" in the disruption the Royal Family is currently experiencing.
Robert Hardman, royal biographer and Daily Mail journalist, said "it shows the depth of the bond between those two brothers" that they responded to this allegation so quickly, and it demonstrates the desire to "avoid any sense of confrontation" at the meeting.
These talks have been described as "royal history in the making", by BBC royal correspondent Jonny Dymond.
Meghan is expected to join the discussion by phone from Canada.
It is hoped the talks will produce a "next step" on the way to defining the couple's new relationship with the Royal Family - in line with the Queen's wish to find a solution within days.
However, our correspondent said a deal "might take time" to be agreed and implemented.
Among the issues likely to be discussed are what funding the duke and duchess will receive, whether they will keep their titles and what royal duties they will continue to carry out.
The group are expected to review a range of possibilities for the Sussexes, taking into account plans outlined by the couple.
The final arrangement could provide a blueprint for future royal generations, including the Duke of Cambridge's younger children Princess Charlotte and Prince Louis.
Monday's gathering at the Queen's estate in Norfolk - being described as the "Sandringham summit" - will be the first time the monarch has spoken face-to-face with Prince Harry since the Sussexes' announcement.
Prince Charles has returned from Oman after attending a condolence ceremony following the death of Sultan Qaboos bin Said Al Said.
Newspaper reports suggested the couple could conduct a potentially damaging television interview if they were unhappy with the outcome of the discussions.
Broadcaster and friend of the couple, Tom Bradby - who made a documentary with them last year - wrote in the Sunday Times: "I have some idea of what might be aired in a full, no-holds-barred, sit-down interview and I don't think it would be pretty."
In Sandringham later the players will receive a series of options, a range of possibilities. These will be based on the stated aims of Prince Harry and Meghan.
Leaving aside the heady brew of contradictions detailed elsewhere, the balancing of these different aims and demands is hard enough.
Money is a big issue. But so will be the status of the court of Prince Harry and Meghan that emerges.
Maybe a deal comes on Monday, Tuesday, Wednesday. But what are not up for negotiation are service and survival. Both must be observed by Prince Harry and Meghan if they are to remain royals.
But will the couple really agree to the restrictions that service and survival demand?
A deal will probably be crafted - however the direction of travel is one way. Prince Harry and Meghan are looking for the exit.