Legislation could be put to a vote in the coming months.
'He fought to make health care a right for all Americans, and we will do that in his honour,' said Senator John Kerry, the 2004 presidential candidate whom Kennedy supported. 'He changed the course of history,' Kerry said.
Earlier Friday, mourners gathered in Boston for the second day to pay their respects to the enduring political icon of the Kennedy dynasty who passed away late Tuesday, aged 77, at his home in Hyannis Port, Massachusetts, after a battle with brain cancer.
More than 50,000 people filed through the Kennedy library to view the casket draped with an American flag and under a US miliary honour guard.
Mass is scheduled to take place at Our Lady of Perpetual Help Basilica in Boston Saturday morning, when President Barack Obama will deliver the eulogy.
Kennedy's body will then be transported to Washington for a late afternoon burial at Arlington National Cemetery, the graveyard near the Pentagon that is the final resting place of many of America's war dead.
Kennedy will be buried not far from the graves of his assassinated brothers - president John F. Kennedy who was gunned in 1963 while in office and Robert Kennedy who was also fatally shot in 1968 while campaigning for the presidency.
At the private Boston memorial, Senator Chris Dodd - Kennedy's closest friend in the Senate - said: 'John Fitzgerald Kennedy inspired our America, Robert Kennedy challenged our America, and Teddy changed our America.'
Despite the scandal that sometimes dogged Kennedy and his family, he was praised for his tireless fight against poverty and for better health care during his 47 years in the Senate and as the backbone of the Democratic Party.