Cabinet minister Penny Mordaunt is among supportive of the plans, having previously lobbied the prime minister to create a new vessel last year.
Prince Philip, who died on Friday at Windsor Castle aged 99, served with the Royal Navy in the Second World War and was Lord High Admiral at the time of his death.
He travelled 70,000 miles on Britannia, including on two round-the-world trips, before it was decommissioned during a ceremony in Portsmouth in 1997.
Ms Mordaunt, Portsmouth North MP and former international development secretary, previously insisted a successor to the royal yacht could be deployed to humanitarian crises, taking the strain from the 'grey hulls’ of Royal Navy warships – while also being used to bring an economic boost for Britain.
Such ships would act as a UK Aid Maritime capability, Ms Mordaunt previously said in a letter to Mr Johnson, and be UK-flagged and controlled.
Speaking to The News today, Ms Mordaunt said: ‘I have long pushed the idea of a vessel that could be used by the UK for a huge variety of roles, from trade, to humanitarian, to education, to research and mariner training, to accommodation for UK staff on particular missions.
‘It would be cheaper to do this than the way we currently procure ships to do a range of tasks. There are many organisations behind this idea and I hope it gains ground. I hope he would have approved of the concept.’
One Cabinet minister told the Sunday Telegraph that the ship could receive backing if it doubled as hospital ship or a training vessel as well as serving the Royal Family.
They said: 'Having a symbol of the nation that can travel the world, be used by the Royal family and have another sensible purpose such as helping young people is a better scheme. It could also be a flagship for reinvigorated British shipbuilding.'
They added that an announcement on such a ship, which would cost around £190m, could be tied to the Queen's diamond jubilee next year.
The ‘Britannia 2.0’ concept vessel, could be built in British shipyards, MPs hope.