Alpine would not confirm details of the test or who was driving last year's car at the Hungaroring circuit near Budapest.
The official Formula One website said de Vries, Doohan and Italian Antonio Giovinazzi (28) were expected to get time at the wheel.
Two-time world champion Fernando Alonso (41) is leaving for Aston Martin at the end of the year, leaving the most desirable remaining vacancy on the grid. Alpine are fourth overall in the Constructor's Championship with six races remaining.
The team had wanted to put Australian reserve Oscar Piastri (21) in the race seat next year but he has signed for McLaren instead.
Frenchman Pierre Gasly (26) then became the favourite but he is contracted for 2023 to Red Bull-owned AlphaTauri, who have said they would be willing to release him if they can find an exciting replacement.
American IndyCar driver Colton Herta (22) was the top target but does not have a super-licence and Formula One's governing FIA is not prepared to make an exception despite the absence of US drivers in the sport.
De Vries, who impressed by finishing ninth on his race debut with Williams as a stand-in for Alex Albon (24) in Italy this month after the Thai developed appendicitis, could be the answer for Alpine and AlphaTauri.
De Vries told Dutch television at the weekend that he recently met Red Bull's motorsport head Helmut Marko in Austria for talks.
"I have been in talks with Williams for a long time... I have been in contact with Alpine since July, and I will test for them in Budapest next week," he said.
"Wherever I can get a permanent seat, I would be very happy with that."
De Vries, the 2021 Formula E champion, and Red Bull's Formula One world champion and runaway leader Max Verstappen are friends as well as compatriots.
That has fuelled speculation that de Vries could end up at AlphaTauri, allowing Gasly to join Alpine.
Academy driver Doohan, the son of motorcycle great Mick, is still a teenager and looks more like a possible reserve for Alpine unless they do a deal with Williams for him to replace Canadian Nicholas Latifi (27).
