"It doesn't solve the problem. But look, we're having those discussions... We're going to continue to meet about this. I know the House leaders are talking about, the Senate leaders are talking about it," said South Dakota Republican John Thune, who chairs the Senate Republican Conference, on "Fox News Sunday".
"Republicans are looking at different options about how best to respond to the president's unilateral action, which many people believe is unconstitutional, unlawful action on this particular issue."
Obama is expected to announce a series of executive actions on immigration issues before the end of the year, Secretary of Homeland Security Jeh Johnson said on Saturday.
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Thune and Republican Representative Tom Cole said Obama's expected use of executive power on the issue was what fueled partisanship and lack of cooperation.
"I think the president wants a fight. I think he's actually trying to bait us into doing some of these extreme things that have been suggested. I don't think we will," Oklahoma Representative Tom Cole said on ABC's "This Week".
Cole said a shutdown was an inappropriate tool and urged a legal challenge to Obama's action.
Democratic Senate Majority Whip Dick Durbin of Illinois said House Speaker John Boehner could move on the immigration bill already passed by the Senate, whose control Republicans gain next year as a result of this month's elections.
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"The message of the last election was, 'solve problems, don't just go to a political standoff, do something,'" Durbin said on CNN's "State of the Union." "If the Republicans fail to do it, then the president will act and I will support it."