Although it had been widely assumed Dovizioso was prepared to commit to Ducati for another two years, recent reports in the Italian media have linked him to a sensational KTM switch.
The 34-year-old's manager Simone Battistella recently declared that the Borgo Panigale marque needed Dovizioso more than Dovizioso needed Ducati.
When asked for his opinion, Brivio said he would pair Dovizioso, who has finished runner-up to Marc Marquez in each of the past three seasons, with a younger rider.
"I think Dovizioso has proved to be a rider who can win races, and for several years now he has been the rider who has been closest to Marquez, so I would keep him as a guarantee for the future," Brivio said in a recent teleconference with the media.
Joan Mir, Alex Rins, Team Suzuki MotoGP
"Joan definitely has the potential to compete with Alex," said Brivio. "One of the things we were looking forward to with great confidence this season was the competitive atmosphere within the team, which both are already starting to feel.
"That's what we missed a little bit last year. In 2018, when Iannone was there, Alex was the up-and-coming rider and wanted to beat him. This internal competition allowed us in the last six or seven races to always be on the podium with at least one of them.
"When one rider was on the podium, the other got angry and tried to do it the next time. I hope this atmosphere will come back in 2020 and I think the conditions are in place."