Dovizioso's future in MotoGP is uncertain after he announced during last month's Austrian Grand Prix weekend that he would be quitting Ducati at the end of the season, bringing to an end an eight-year partnership that has yielded 14 victories and three runner-up spots in the championship.
At the time, Dovizioso admitted he had no "plan B" for his future, and the MotoGP points leader still has no update to offer.
He has been linked to ride with Aprilia - which has also been speaking to Cal Crutchlow - but CEO Massimo Rivola conceded Aprilia likely couldn't afford Dovizioso.
Dovizioso is open to taking a sabbatical in 2021, with a test rider role likely to involve wildcard appearances - though only Aprilia would be able to offer him more than three owing to the concession rules.
With Jorge Lorenzo's role as Yamaha test rider not secured beyond this year, Rossi was asked if he would like Dovizioso as an alternative.
"About Dovi [as Yamaha test rider], for sure," he replied. "I'd sign now to have Dovi as a test rider.
"But I think, and I hope that he will find a bike for racing, because I think he is young and fast to race."
After his disastrous 2019 campaign with Honda, Lorenzo retired from racing at the end of year and returned to Yamaha as a test rider.
However, he has only completed two days on a 2019-spec M1 so far this year - one at the Sepang shakedown, and once during the official Sepang test.
He is due to ride the Yamaha at the Algarve circuit in the coming weeks as part of a Michelin tyre test ahead of November's season-ending Portuguese Grand Prix.
In an exclusive interview with Motorsport.com this week, Lorenzo said his "priority" was to renew his deal with Yamaha for next year - though confirmed rival brands have spoken to him.
Rossi and Fabio Quartararo questioned Yamaha's decision not to test with Lorenzo at Misano in June to try the new surface when its rivals did.
Yamaha later revealed this was a result of its test team, which is largely based in Japan, being unable to get to Europe owing to COVID-19 restrictions.
Yamaha did have a European-based test team headed by Jonas Folger in 2019 as well as its Japanese arm, but it was thought better to combine the two teams going forward.
Alex Rins was also asked about the possibility of Dovizioso being signed by Suzuki to bolster its test team for 2021.
He believes the Italian would be an "interesting option" because of his experience of Ducati, Yamaha and Honda machinery in the premier class.
"For sure, will be an interesting option because to have Sylvain [Guintoli] and Dovizioso will be perfect, because Dovi has a lot of experience and he knows the rest of the bikes," Rins said.