It also seeks information about US "political and military objectives regarding Libya."
Boehner offered his resolution amid growing support, even among Republicans, for a separate measures that sought a withdrawal of US forces involved in the NATO mission within 15 days of passage.
That resolution, proposed by antiwar Democrat Dennis Kucinich, failed 265-148, with 87 Republicans voting in favor.
The White House called the resolutions "unhelpful and unnecessary."
Boehner noted that Congress has the constitutional authority to declare war and that presidential authority is limited by the War Powers Resolution of 1973.
Several lawmakers from both parties have recently contested Obama's continuation of military operations in Libya under the War Powers Resolution, which gives the president 60 days to seek congressional approval for military action.
That deadline, which has been routinely ignored by several presidents since the law was passed in 1973, was reached on May 20.
Obama has argued that the limited nature of American intervention in Libya is not the kind of military operation envisioned by the law.
"We have continued to consult with Congress all along," White House Deputy Press Secretary Josh Earnest told reporters aboard Air Force One ahead of the votes.
"In just the last week there have been three separate congressional briefings."
Some senators are also complaining that they have been ignored. The Senate is expected to soon vote on a resolution of support for a "limited" US intervention, sponsored by Republican John McCain and Democrat John Kerry.
Kucinich had argued that the Libya intervention was a matter for Congress.
"Our loyalty to NATO and to our president, regardless of party affiliation, does not trump our loyalty to the United States Constitution," the ardent critic of US intervention overseas wrote Thursday in a letter to his colleagues.
Republican Representative Ileana Ros-Lehtinen, who chairs the House Foreign Affairs Committee, said "members of both sides of the aisle are increasingly frustrated."
"If this clear warning doesn't get the attention of the White House, then more forceful action may be inevitable," she added.
Boehner said Congress would consider other measures in "the weeks to come."
"Clearly, there's information that we want from the administration that we ask for in this resolution, and it's information that we expect to get. But there is not any question in my mind that Congress is going to take further action in the weeks to come," he said.
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