Kennedy had been on several flights with Wellstone on the twin-engine turboprop but did not accompany him on the fatal flight as Wellstone was traveling to a funeral and not a campaign event.
Wellstone, his wife, daughter, three staffers and the two pilots were all killed in the crash.
The Cape Cod Times first reported this weekend's medical emergency and has posted a photo of Kennedy being taken to a MediFlight on a stretcher.
A Democratic Senate source told ABC News, "The situation is serious. We just don't know how serious."
The Kennedy family was set to host the annual Best Buddies Challenge event Saturday evening at the Kennedy family compound.
Even though he is the third-longest serving senator in the history of the body -- with 45 years and two months in office, behind only current Sen. Robert Byrd of West Virginia and Strom Thurmond of South Carolina -- Kennedy is one of the hardest-working members of Congress.
As the chair of the Health Education and Labor Committee, he was the Democrat who ran the Senate floor last week when senators were considering a bill to give collective bargaining rights to all police and firemen. He was also the chief Democrat who fought for a recently passed student loan improvements bill.
Kennedy chaired a widely covered hearing on May 12 about how to fight cancer. Among the people who testified during that hearing were Lance Armstrong, Elizabeth Edwards and Steve Case.
Kennedy was also set to play a pitvotal role for Democrats as they consider the supplemental war funding bill next week. Kennedy was one of the loudest critics of the war in Iraq from its outset.
Before his illness this weekend, Kennedy was apparently working; his office released a statement in his name, congratulating Benjamin Jealous on his election as head of the NAACP.
Kennedy and his brother Robert, who was a senator from New York, are the only two people to serve in the Senate at the same time as siblings.
Sen. Barack Obama, D-Ill., who received an endorsement from Kennedy and his niece Caroline Kennedy before the Super Tuesday contests in February, told reporters he has "been in contact with the [Kennedy] family."
"They are in our thoughts and prayers," Obama said. "As I have said many times before, Ted Kennedy is a giant in American political history. He has done more for health care of others than just about anybody in history, and so, we are going to be rooting for him. And I insist on being optimistic about how it's going to turn out."
Sen. John McCain, R-Ariz., the presumptive Republican presidential nominee, also released a statement on Kennedy, saying, "I was very sorry to hear that Sen. Kennedy has taken ill, and like millions of Americans, Cindy and I anxiously await word of his condition. Sen. Kennedy's role in the U.S. Senate cannot be overstated. He is a legendary lawmaker, and I have the highest respect for him."
Sen. Hillary Clinton, D-N.Y., campaigning in Oregon, said, "Our thoughts and prayers are with Ted Kennedy and his family today. We wish him well, and a quick recovery."
Sen. John Kerry, Massachusetts' other senator, who was at the hospital with Kennedy and other members of his family, released a statement saying, "Ted Kennedy is beloved and respected on both sides of the aisle in the Senate, in which he's been a giant for close to half a century, a legend in Massachusetts, and a dear friend to me and [Kerry's wife] Teresa.