The trips will involve two years of international travel aimed at "extending a hand of friendship and support", royal sources told the Mirror.
New Zealand, Australia and a number of Commonwealth realms in the Caribbean are top of the monarch's priority list, the Mirror reported.
The Prince and Princess of Wales were also understood to be aboard in a "major support role" as the royals start a new era after Queen Elizabeth's passing.
The King is also expected to undertake a number of solo trips, both at home and overseas.
This would allow the Queen Consort to undertake more engagements for her personal projects, the Mirror said.
A royal source told the Mirror: "The King and his family are keen to hit the ground running in these crucial first few months and years of his reign.
"He certainly wants to carry on the long-held mantra from his late mother of being seen to be believed, and is very keen to get out and meet as many people as possible."
In the 1970s, the Queen crammed in 52 Commonwealth visits and 21 trips to other countries to mark her 25th jubilee. Sources told the Mirror the King could undertake just as many trips.
The King has reportedly opened discussions with his chief advisers over visits to major British realms including Australia and Canada "as soon as possible".
He has also raised the importance of visiting the 14 other nations where he is king.
Government officials in New South Wales this week revealed plans for a "much-anticipated" royal tour in 2024 to coincide with the 200th anniversary of democracy in Australia.
However, sources suggested Prince William and Kate had also previously discussed plans for a tour Downunder in spring 2023.
