Charting the Fife town's medieval history, they are to be housed in the existing Carnegie library and a disused bank.
Fife Council has pledged £6.8m and the project has a first round pass for £2.8m from the Heritage Lottery Fund.
A first round pass means it meets the HLF's criteria for funding before detailed plans have been submitted.
The project now has up to two years to submit fully developed proposals to compete for a firm award.
The buildings would be redeveloped and a new extension added to create a huge development including a museum, art gallery, archive, library and local history services.
Colin McLean, head of the HLF in Scotland, said: "Museums, galleries and libraries are extremely valuable community assets.
"They make a great contribution to people's education, self-identity and enjoyment.
"They keep our history safe for future generations and they impact on the local economy, often as the cornerstone of an area's tourism industry.
"We are delighted to be able to give Dunfermline our support at this stage."
Jim Tolson, MSP for Dunfermline and West Fife, said he was "overjoyed" with the backing.
"HLF's support for this project is crucial to the long-awaited addition of a museum and art gallery in Dunfermline which will benefit both local people and visitors for generations to come," he added.
Brian Goodall, chairman of Fife Council's housing and communities committee, said: "The Heritage Lottery Fund's decision to support the development of Dunfermline's new flagship museum is a tremendous vote of confidence in the city. We are delighted with the news.
"Fife Council has itself made a major commitment of £6.8m to delivering this project as part of the council's drive to improve leisure and cultural opportunities for all."

The 21-year-old woman, who police said was a sex worker, was found early on Wednesday in Springwell Road, off Water Lane, in Holbeck.
She was taken to Leeds General Infirmary for treatment, but died a short time later.
A 38-year-old man was later arrested and questioned by officers investigating the death.
West Yorkshire Police said the woman's identity was known, but her family had not yet been informed.
Det Supt Simon Atkinson said they wanted to hear from anyone who was in the Springwell Road area between 20:00 GMT on Tuesday and 01:30 on Wednesday.
He added: "We can confirm that the victim was a sex worker and the incident has taken place in the managed area where sex workers operate within a defined area during set hours.
"We are therefore appealing to anyone involved in any aspect of sex work in the managed area in Holbeck to let us know anything they have seen or heard that could assist the investigation."
A new museum and art gallery are set to be built in Fife under multi-million pound plans for Dunfermline.
[NEXT_CONCEPT]
A murder inquiry has begun after the death of a woman found seriously injured in Leeds.