She said: 'The Queen always has this wonderful, extended stay for her grandchildren and her great-grandchildren. There's a big sleepover at Balmoral, they were invited to go to that.
'They were also invited to go to Birkhall with the Prince of Wales and thus far they haven't done it.'
The US-based couple, who are thought to have returned to Windsor over the weekend, are on a whirlwind European trip, having visited Dusseldorf today for an Invictus Games event, ahead of attending the WellChild Awards in the UK on Thursday.
Katie said: 'I know from sources very close to the Prince of Wales that he is prepared to do anything it will take to get this relationship back on track, I think not least because he wants to as a father.
'He wants to see his children as a grandfather, but I think he's acutely aware that if there is this rift that threatens to overshadow his future reign as well. There are other issues at play here.'
The Sussexes are believed to have arrived at Frogmore Cottage in Windsor on Saturday ahead of their European tour.
The property is located just 380 feet away from the Cambridges' new four-bedroom Adelaide Cottage, where Prince William and Kate spent the weekend with their children.
However, while Harry and Meghan spent time less than half a mile from the Cambridges, the families are not expected to meet, in a decision that underlines the deep rift between the brothers.
Katie said: 'I'm told I don't think William can forgive his brother for what he's done and how he's done it, in terms of not just leaving the royal family, throwing them under a bus and his treatment of the royal family today.
She continued: ''Of course, we aren't getting any official comment from the palace because if it happens, it'll be a private meeting.
'But everything I'm hearing is that there is still very much a rift and let's not forget there is Harry's autobiography to come. They don't want to perhaps see him for risk of something being leaked.
The royal expert also revealed that she spoke to a person helping to organise last night's One Young World summit in Manchester, in which both Harry and Meghan attended.
'I spoke to someone who was at the One Young World event, who was helping to organise it, who assured me there was no Netflix camera at this event,' said Katie.
'She says it went swimmingly at the event last night in Manchester, but I think that there's constant awareness in the royal family that there's going to be an autobiography.