WASHINGTON — After millions of Americans vote next week, it’s possible that one or two men will decide which party controls the Senate.

One is Sen. Angus King, a Maine independent who generally “caucuses” — or cooperates — with Democrats but says he might switch to the Republicans. The other is Greg Orman of Kansas, an independent candidate trying to oust Republican Sen. Pat Roberts.

If he wins, Orman says, he would caucus with whichever party holds the majority when the new Congress convenes in January. He has not said, however, what he would do if he could decide, by himself, which party that will be.

That could happen if Republicans win 50 seats and Democrats control 49 seats (including King’s and that of another independent, Vermont’s Bernie Sanders, who also caucuses with Democrats). If Orman sided with the Democrats, they would control a 50-50 Senate, thanks to Vice President Joe Biden’s ability to break tie votes.

But if Orman sided with the Republicans, they would hold a 51-49 majority.

King, who’s not up for election this year, could complicate things even further. In the same scenario as above — a 50-49 GOP advantage with Orman undeclared — King could put the Republicans in charge by switching to their caucus. Orman’s decision, either way, would not change that.

In a different scenario — in which Democrats held a 50-49 edge awaiting Orman’s decision — both he and King could choose to caucus with Republicans, giving the GOP a 51-49 majority.

Congressional insiders see that as unlikely. First, Orman would have to win an election that many Republicans believe Roberts will survive. Then, all the other races would have to produce a 50-49 split, awaiting Orman’s declaration. And King would have to switch party leanings and hand minority status to Democrats, with whom he generally seems comfortable.

“Changing who you caucus with is akin to changing parties,” said Antonia Ferrier, a former GOP House and Senate aide. “It is no small thing, is deeply personal and exceedingly rare.”

King occasionally hints he would consider such a switch. He told reporters in April that he will do what is “best for Maine” in the next Congress. He made similar remarks last week, stirring a Senate campaign pot already about to boil.

Currently, Democrats control 55 Senate seats to the Republicans’ 45. Republicans need to gain six net seats to seize the majority.

King repeatedly calls for more bipartisanship and cooperation in government. He has urged Orman and another third-party candidate — former Sen. Larry Pressler of South Dakota — to refuse to say which party they’d caucus with in Washington. Pressler’s chances seem to be fading, but Orman is giving Roberts a tough fight.

People close to King say they’d be surprised if he joined the Republicans’ caucus, especially if doing so handed the GOP the majority. He supported President Barack Obama’s 2012 re-election, and his voting record leans more Democratic than Republican.

But King has endorsed some Republicans for re-election, including Sens. Susan Collins of Maine and Lamar Alexander of Tennessee. In Maine, where Republican Gov. Paul LePage seeks re-election, King originally endorsed independent candidate Eliot Cutler but then switched to Democrat Mike Michaud.

Even a remote hint of a caucus switch creates buzz in Senate circles.

“If — and it’s a big if — control of the Senate is determined by two independents, then there well could be some quiet outreach by both sides over which party they should caucus with,” Ferrier said.

Inducements from party leaders might include promises of plum committee assignments. Former Democratic Senate aide Jim Manley said party leaders also might try to address King’s concerns with overtures such as pledging more bipartisanship or a more open process for legislative amendments. Manley said the entire party caucus would weigh in on such matters.

Senate leaders have learned not to overdo favors to lawmakers withholding their votes.

In early 2010, then-Democratic Sen. Ben Nelson of Nebraska secured a promise of $100 million in Medicaid funding in return for his crucial vote for Obama’s health care overhaul. Critics denounced the “Cornhusker kickback,” which eventually was dropped.

In 2001, veteran Sen. Jim Jeffords of Vermont left the Republican Party to become an independent who caucused with Democrats, handing them control of the Senate. Jeffords surrendered his chairmanship of the Health, Education, Labor and Pensions Committee and accepted the Democrats’ offer to chair the Environment and Public Works Committee.

Senate aides say it would be virtually impossible for either party to offer a committee chairmanship to King, who has been a senator only two years.