"Carrie is in stable condition," Reynolds tweeted
. "If there is a change, we will share it. For all her fans & friends. I thank you for your prayers & good wishes."
The United Airlines flight was on approach to Los Angeles on Friday when the 60-year-old actress who played Princess Leia in "Star Wars" became ill.
Fisher was transported to a hospital from Los Angeles International Airport, the source said.
The Los Angeles Fire Department previously said it responded to a cardiac arrest at the United Airlines gate where the London flight arrived.
"LAFD responded to LAX Gate 74 for patient on inbound flight in cardiac arrest," LAFD Spokesman Erik Scott told CNN. "Paramedics standing by upon arrival provided advanced life support and aggressively treated and transported patient to local hospital."
Celebrities aboard the same flight took to Twitter to express their concerns.
Comedian Brad Gage tweeted: "I sat in front of Carrie Fisher on our flight from London and she was just taken off the plane by EMTs."
YouTube personality Anna Akana tweeted that the actress "stopped breathing" on the flight. Akana thanked the doctor and nurse who came to Fisher's aid.
The on-screen princess
Fisher was in London filming the latest season of Amazon's "Catastrophe." Just days ago, she tweeted a photo of herself and actress Sharon Horgan on set.
Born in Beverly Hills to entertainers Debbie Reynolds and Eddie Fisher, Carrie Fisher got her start in Hollywood in 1975 in the movie "Shampoo," starring Warren Beatty.
Best known for her portrayal of Leia in the "Star Wars" movie franchise, Fisher went on to act in plays, television and blockbuster films and has written a number of best-selling books.
Fisher says in her new memoir, "The Princess Diarist," that she and Harrison Ford had an affair
on the set of the original "Star Wars," which hit theaters in 1977. Ford, who was married at the time, has not confirmed or denied it.
Ford and Fisher reprised their roles last year in "Star Wars: The Force Awakens
Her work has earned Fisher two Emmy Award nominations.
Fisher has been a longtime advocate for mental health awareness, drawing from her own struggles with bipolar disorder.