Donald has guided Europe to consecutive victories at the event following their triumph at Bethpage Black in September and Fleetwood has stressed that he and his team-mates want him to lead the charge again in two years time.
The English star told the Sky Sports Golf podcast: "I think all of us, as players and as a team, would love Luke to stay on.
"He has been an unbelievable captain and an unbelievable leader.
"I would class Luke as a very close friend and to watch how he has grown as a leader over this period, all the way through his captaincy in Rome to now, to watch him grow into it has been amazing.
"I think we would love him to stay on and continue. I feel like the story has got a little bit to go, so I would love him to do that, but it's completely his choice.
"With the captain he has been, he has earned the right to decide what he wants to do. We will see. He has been amazing and we will wait to see what Luke Donald wants to do next."
Fleetwood was Europe's top scorer during their historic 15-13 win over the USA in New York and is looking forward to watching his exploits back over Christmas.
The world number four said: "I would say we're all proud to have gone there as a team, and as a family, and represented Team Europe the way that we did.
"For all of us, that will be some of the most memorable moments of our career – whatever happens in the future.
"I actually love the Christmas period because that's when the programmes come out and the re-runs of the Ryder Cup come back, so I'll look forward to watching them back!"