By Susan Cornwell

WASHINGTON (Reuters) - A senior U.S. Senate Republican warned his party on Thursday against simultaneously overhauling Medicare and the Obamacare health insurance program, saying this would be "biting off more than you can chew."

The cautionary comments from Senator Lamar Alexander came after House Speaker Paul Ryan, long an advocate of privatize Medicare, said Republican lawmakers would be discussing reforms of the health insurance program for the elderly with President-elect Donald Trump's administration.

Republicans won the White House as well as keeping their majorities in both houses of Congress in elections last month, and are now busy preparing an agenda for next year after the new Congress is sworn in on Jan. 3 and Trump takes office Jan. 20.

"Well, of course it’s up to the speaker and Senator (Mitch) McConnell (the majority leader) what our agenda is, but my advice to them would be, save Medicare for another day," Alexander, chairman of the Senate's health, education, labor and pensions committee, told reporters in a Capitol hallway.

Medicare serves more than 50 million Americans who are elderly or disabled. More than one-third of them are in Medicare Advantage plans run by private insurers like UnitedHealth Group Inc. The balance receive medical benefits directly from Medicare on a traditional fee-for-service basis.

The government has been trying to cut spending on the program, which rises each year with the cost of medical services and drugs.

"We want to begin immediately to repeal Obamacare," Alexander said. "Trying to deal with the solvency issues in Medicare at the same time falls into the category of biting off more than you can chew ... a little humility here would be in order, we can’t do everything at once and we shouldn’t try."

Ryan earlier Thursday said that Medicare was on a path to going bankrupt around 2028, and needed reform, a repeat of his long-standing stance. He wants to convert the fee-for-service program into a system of subsidies for seniors, to buy coverage from private insurers or a scaled-back Medicare.

Democrats including Senator Chuck Schumer have seized on the fact that Trump's nominee to head the Health and Human Services department - Representative Tom Price - supports major changes to Medicare, to warn Republicans against privatizing the program.

Alexander also said that while Republicans will move quickly to repeal Obamacare, the repeal probably would not take effect for two or three years, during which time lawmakers can work on a replacement plan.

(Reporting by Susan Cornwell; Editing by Tom Brown)