Kate and William would spend more time at Balmoral and strengthen ties with their former university town of St Andrews if the proposals go ahead, according to the Sunday Times.
The report comes just days after the couple’s tour of Scotland in which William said the country was “so important” to himself and wife.
The couple sat down for talks with the former prime minister and his wife Sarah at the Queen’s official Edinburgh residence.
William later gave a highly personal farewell speech, describing how Scotland has “shaped” him and praising its people and values.
Soon after the Holyrood election Mr Brown announced his Our Scottish Future think tank which will become a “campaigning movement” seeking to appeal to “middle Scotland”, those who are not entrenched in their positions on the union or independence.
A YouGov poll in Scotland in April found that 69 per cent of voters had a positive view of William, with 17 per cent having a negative view. By contrast, YouGov’s most recent survey found that just 22 per cent of Scots had a favourable view of Boris Johnson, while 71 per cent had an unfavourable view.
The argument over Scottish independence has intensified after the Scottish National Party’s landslide victory in the Holyrood election in May, which also produced the largest pro-independence majority in the Parliament in the history of devolution.
Mr Johnson has since stood by his pre-election position, saying the focus should be on the recovery from Covid-19 and not on another independence referendum.
Details of the plans for William emerged after the prime minister hosted a meeting on Thursday with the devolved administrations designed to strengthen links between England, Northern Ireland Scotland, Wale.
Among the plans discussed were more money being invested in transport links between the four nations and British diplomats being told to change the way they speak about the UK, referring to it as one country rather than talking about “the four nations of the UK”..
A survey in March found that 39 per cent of Scots favour retaining the royal family if the UK breaks up while 39 per cent said a Scottish republic should be created and 22 per cent said they did not know.