Their caution is evident when they try to explain their positions on it.
"If it leads to citizenship as a reward for some kind of illegal entry, I think it could be argued" to be amnesty, said Senate Republican leader Mitch McConnell, who normally is one of Obama's most plainspoken critics.
Sen. Saxby Chambliss of Georgia said the president's announcement last Friday could be called "amnesty light," if not the real thing.
"It all depends on how you define `amnesty,' and I'm not going to get into that debate," added Arizona Sen. Jon Kyl, second-ranking party leader and another frequent critic on Obama.
"I wouldn't use the A word to describe this. ... That's a word that gets used a lot," said Sen. John Cornyn, the Texas lawmaker who is in charge of the party's efforts to win control of the Senate this fall and has already seen internal divisions develop between his party's establishment and the lawmakers more aligned with the tea party.
These days, at least, amnesty is most often used by Republicans wanting to accuse Democrats of being soft on illegal immigration, much of which involves Hispanics who make their way across the border from Mexico.
In his announcement, Obama said the government will no longer seek to deport illegal immigrants under the age of 30 who were brought to the United States before they turned 16 and have been in the country for at least five continuous years. They also must have no criminal history and either have graduated from a U.S. high school, have earned a GED diploma or have served in the military.
"Let's be clear, this is not amnesty, this is not immunity, this is not a path to citizenship, this is not a permanent fix," Obama said from the White House Rose Garden. "This is the right thing to do." Administration officials said about 800,000 individuals could be affected.
While the White House and Democrats in Congress say the president's action was modeled on the DREAM Act, that measure includes a path to citizenship, a critical political distinction.
Romney, who has vowed to veto the DREAM Act, has recently softened his rhetoric on immigration as he works to cut into Obama's support among Hispanic voters. So far, he has not said if he supports or opposes what the president did.
McConnell and others said they want to hear what the former Massachusetts governor is going to say on the subject in a speech later this week. Romney is "the leader of our party" at least until the November election and perhaps well afterwards, McConnell said.
Whatever Romney says, Republicans aligned with the tea party are more outspoken than the party's establishment on the issue.
"This is amnesty," Sen. Jim DeMint, of South Carolina, a favorite of tea party Republicans, said last week in a quick reaction to Obama's announcement.
"Congressman (Jeff) Flake and President Obama, advocates for amnesty," Wil Cardon alleged in one emailed statement, issued as part of his uphill battle for the GOP nomination to the Senate in Arizona.
Flake's campaign did not immediate answer a request for a response.
Rep. Steve King, R-Iowa, one of the party's more outspoken lawmakers in the immigration issue, has announced he intends to file suit challenging Obama's authority to bypass Congress and change the policy on his own.
Also in the House, Rep. Ben Quayle, R-Ariz., has introduced legislation to block Obama from implementing his own policy. In some regards, his state serves as a microcosm of the broader debate inside the party.
In addition to Kyl's comments, Sen. John McCain, R-Ariz., told reporters during the day he neither wanted to "repeal it or not repeal" what Obama announced.
Like many other Republicans, McCain said Obama was now doing "something that he said he didn't have constitutional authority to do a year ago." Also like other members of his party, he said the president's announcement had complicated efforts to work out a comprehensive solution to a vexing problem.
House Speaker John Boehner told reporters that because the economy is so weak, Obama has "turned to the politics of envy and division, which I don't think the American people are going to accept."
Boehner said Obama's announcement "puts everyone in a difficult position," including young illegal immigrants who "from no fault of their own" are in the United States, as well as lawmakers seeking legislation to tackle more than just the problem of younger illegal immigrants brought to the United States as children.
Democrats have applauded Obama's announcement, many of them adding that he acted after Republicans refused to permit a more wide-ranging immigration measure advance in Congress.