The settlement stems from VW's admission in September that it intentionally misled regulators by installing secret software that allowed US vehicles to emit up to 40 times legally allowable pollution. It covers 475,000 2.0-liter vehicles.
The Justice Department filed a proposed consent decree that confirms VW will set aside $10.033 billion to cover buybacks and fixes, $2 billion to invest in green energy funds and $2.7 billion to offset diesel emissions.
Shares of VW were up 4.6 percent at 111 euros.
A source briefed on the matter said VW would announce a separate settlement with at least 44 US states, the District of Columbia and Puerto Rico that will cost at least $600 million.
The actual amount VW will spend on buybacks could be significantly less if regulators approve fixes and owners opt to get vehicles repaired.
Most owners will get at least $5,100 in compensation in addition to the pre-scandal value of the cars and up to $10,000, documents filed on Tuesday said.
Volkswagen expects to begin buying back vehicles in October, when a US judge is expected to give final approval to the settlement and is to start proposing fixes in November. Some vehicles will require significant mechanical fixes.
VW cannot resell or export the vehicles bought back unless the US Environmental Protection Agency approves a fix, the documents said. Volkswagen must repair or buy back 85 percent of the 475,000 vehicles by June 2019 or face penalties of $100 million for every percentage point it falls below that figure.
The settlement does not cover fees for the lawyers of owners suing VW or address 80,000 larger polluting 3.0 liter Porsche, Audi and VW diesel cars.
Also to be decided later is the amount of any civil fine VW faces under the US Clean Air Act for emissions violations.