The current age for Medicare eligibility is 65 and has been 65 ever since the program was created. However, there are currently debates over raising it in two month increments each year until 2027, when the eligibility age will reach 67. According to a report released by the Congressional Budget Office (which can be found here) this would reduce the federal budget deficit by $148 billion over the next 10 years. Compared to the $15 trillion of government debt, it's not a whole lot, but it is definitely quite a bit of money.
However, the cost for the 65- and 66-year-olds, their employers, and the states would be $220 billion, quite a bit more than the savings by the government. Additionally, an estimated 25% of 65- and 66-year-olds would find themselves uninsured. So is this really a good idea?
There are arguments for both sides. As seen, the total costs in the long run would be higher, but it would lower the federal budget deficit, which at $15 trillion needs to happen. It could also encourage more people to work and to keep their job until they're older so that they still have health insurance, however the CBO expects this effect to be small. Still, this can contribute to a more thriving economy, but would also give college graduates a harder time finding a job as people are taking longer to retire and less replacements are needed in the work force. Medicare premiums might go up because the program would lose its healthiest beneficiares (the 65- and 66-year-olds) and private insurers would suddenly be getting more older beneficiaries, increasing their premiums as well. And not everyone would be able to turn to private insurance; an estimated 5 percent would become uninsured.
There are obviously other effects and considerations as well, including changing the early eligibility age and the full retirment age. I really recommend that you read the full report from the Congressional Budget Office yourself as it includes a lot of good information as well as different scenarios. I do, however, leave you with this thought, should we decrease the federal budget deficit or should we guarantee that our 65- and 66-year-olds are insured? Both decisions affect our future.