Campaigners have also demanded assurances that the Elgin Marbles will not be returned to Greece.
Sir John Hayes, chairman of the Common Sense Group of Conservative MPs, has written to Mr Osborne, saying he hopes his appointment will guarantee the museum "does not pander to the woke agenda which has bedevilled other great institutions".
He said: "Those on the Left [regularly] demand our institutions apologise for historic links to the Empire, which was, in so many ways, a force for good. These, and other, left-wing misconceptions are sufficient in their eyes to legitimise all kinds of cultural distortion and destruction.
"We hope your appointment will ensure such a woke agenda is vigorously resisted in all its facets at the museum.
"We would go further and ask that you promote the museum as a patriotic enterprise which celebrates Britain's glorious history rather than apologises for it."
Sir John also expressed concern at "suggestions famous artefacts such as the Elgin Marbles, or items from the Empire, might be given away".
The marbles, also known as the Parthenon Sculptures, were removed from Athens by Lord Elgin, British ambassador to the Ottoman Empire, between 1801 and 1805.
In 1983, Greece made a formal request for their return.
This year the museum's current chairman, Sir Richard Lambert, said the trustees had "no intention of removing controversial objects from public display" but would "seek to contextualise them in a way that enables the public to learn about them in their entirety, however challenging this may be".
The museum's website says "its history and collection are shaped by empire and the colonial exploitation of people and resources". It says often objects were "collected first by individuals, organisations or companies, passing through several owners before coming to the Museum".
Key contested items include the Benin Bronzes, from the West African Kingdom of Benin.
Mr Osborne will join the museum board on September 1.