Teenager Doohan in the frame for Alpine F1 seat
Doohan is doing well in his rookie Formula Two championship season and is a member of the Alpine driver academy.
Rossi told the website that Doohan was a consideration, even if fourth-placed Alpine would prefer someone more experienced who would be able to score big points immediately.
"We believe in him, he’s shown a lot of potential, especially lately - talk about being present at the right moment," Rossi said.
"He’s a great person, he is fast, his father is also a champion I admire and a person I admire. One way or another we will try and put him in an F1 car at some point."
Mick Doohan was 500cc world champion, before that class became MotoGP, from 1994-98.
Rossi said, however, that Alpine were "once bitten, twice shy" after the recent contract dispute with Australian reserve driver and 2021 Formula Two champion Oscar Piastri (21) who has signed for McLaren for 2023.
Alpine had spent heavily on preparing Piastri and had planned to loan him to Williams, in the same way Mercedes did with George Russell (24).
"We will honour our commitment to all drivers in our academy, but we’re wondering whether or not to continue," said Rossi.
"We are torn apart. We believe in the value of the system but if we’re not protected, is it worth it?...We really wonder. It’s a big disappointment. Perhaps a big reality check."
Experienced French driver and race winner Pierre Gasly (26) is seen as Alpine's first choice to replace double world champion Alonso, who is moving to Aston Martin next year, alongside Esteban Ocon (25).
Gasly's move from Red Bull-owned AlphaTauri depends on Red Bull being allowed to sign American IndyCar driver Colton Herta (22), who does not have enough points for a super-licence and would need special permission from the FIA.
Indications from the governing body are that they will stick to the rules.
"I think if it's not Colton Herta, then Pierre Gasly will stay and nothing will change," AlphaTauri boss Franz Tost told reporters.