"Yamaha and Rossi have mutually agreed that the personal decision whether the Italian will remain an active rider in the MotoGP World Championship in 2021 will be taken mid-2020," they said in a statement.
Rossi has been with the team for 15 years, and Yamaha were quick to confirm that they would provide "full support and total focus" for the coming season prior to any decision about the future.
Quartararo, 20, will remain with the Petronas Yamaha non-works team for one more season before stepping up alongside Spaniard Maverick Vinales, whose contract was extended on Tuesday, at the factory outfit.
Rossi, 40, will be starting his 25th season in grand prix motorcycle racing in March and said he wanted time to decide what came next.
"Yamaha asked me at the beginning of the year to make a decision regarding my future," said the nine times world champion in a team statement.
"It is clear that after the last technical changes and with the arrival of my new crew chief, my first goal is to be competitive this year and to continue my career as a MotoGP rider also in 2021.
"Before doing so, I need to have some answers that only the track and the first few races can give me."
Yamaha team manager Lin Jarvis said the Japanese manufacturer respected Rossi's position but had to plan for the future with hot competition among teams for rising stars like Quartararo.
"It's a weird sensation to start a season knowing that Vale will not be in the factory team in 2021, but Yamaha will still be there for Valentino, whatever he may decide for the future," said Jarvis.
"If he feels confident and continues to race, we will provide a factory-spec YZR-M1 bike and full engineering support."
Jarvis said that if Rossi decided to retire, Yamaha would expand their collaboration off the track in an ambassadorial role and with young rider training programmes.
Only Italian Giacomo Agostini and Spaniard Angel Nieto have won more championships than Rossi, the most successful Yamaha rider of all time.
Rossi has started 402 races, with 89 wins in the top category.
Quartararo had an impressive rookie season in 2019 with six pole positions and seven podium finishes, including five second places, and has signed for 2021 and 2022.
"Now I have a clear plan for the next three years and I‘m really happy," said the Nice-born rider.