Obama pledged to bring anyone responsible for Kayla Mueller's captivity and death to justice "no matter how long it takes."
Of immediate concern was a legislative struggle " the search for a compromise that could satisfy Democrats who oppose the use of American ground forces in the fight against IS, and Republicans who favor at least leaving the possibility open.
Menendez, in describing the White House's opaque formulation, said it remained subject to modification.
One influential Republican, Sen. John McCain of Arizona, said it was "bizarre" for Obama to be asking lawmakers to limit his own power as commander in chief.
A senior Democrat, Sen. Barbara Mikulski of Maryland, said she has significant questions about the president's proposal. "I don't know what the word 'enduring' means. I am very apprehensive about a vague, foggy word," she said.
House Democratic Leader Nancy Pelosi was more emphatic after a separate meeting on the other side of the Capitol. "I'm not for boots on the ground," she said, referring to U.S. combat forces.
Menendez also said it was not yet clear if the proposal would cancel a 2001 authorization for the use of force that Congress approved shortly after the terrorist attacks of Sept. 11, 2001.
Republicans control both houses of Congress, and presidents generally court bipartisan support for legislation of the type Obama now seeks.
Several other lawmakers who were briefed in earlier meetings, said the president would likely seek legislation targeted exclusively against the fighters seeking establishment of an Islamic state, wherever they are and whatever name they use.
Apart from the midday meeting with Democrats in the Capitol attended by White House Chief of Staff Denis McDonough, some Republicans expressed concern with other elements of the administration's emerging proposal.
Sen. Lindsey Graham, a Republican, said administration officials had told him it would not provide for the protection of U.S.-trained Syrian rebel troops on the ground in the event of an air attack by Syrian forces loyal to President Bashar Assad.
"It's an unsound military strategy. I think it's immoral if the authorization doesn't allow for us to counter Assad's air power," he said.
There was little evident dispute in Congress that new legislation was needed, both to replace outdated authorization and also to underscore a bipartisan desire to defeat the terrorists seeking an Islamic state.
Among members of Obama's Democratic party, Rep. Steny Hoyer said during the day that some rank-and-file lawmakers want to set geographic limits and restrict the types of forces that can be used.
"They want some time limit so we can reconsider at some point in time, whether it's 24 months, 36 months, 48 months," he said at a news conference.
Many Republicans have said they prefer legislation that at least permits the use of ground troops if Obama decides they may be necessary. Some, including Sen. John McCain, have gone further, saying ground troops are needed if the Islamic State fighters are to be defeated.
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Associated Press writers Laurie Kellman and Alan Fram contributed to this report.
