WASHINGTON — The House will vote on Wednesday on whether to grant President Obama the authority to train and equip Syrian rebels to battle the Islamic State, but Republicans will insist on a detailed accounting of how the program fits into a broader strategy to defeat the militants, Republican leaders said Tuesday.

House Republican leaders plan to offer the legislation as an amendment to a broader bill to keep the government funded into December and the Export-Import Bank open through June. The Senate would try to pass the bill by the end of the week.

The House speaker, John A. Boehner of Ohio, walking a line between House Republicans who want to do more to confront ISIS and those who do not want to give the president what he wants, said Congress could weigh back in later this year, if Mr. Obama requests broader authority to expand military operations.

Using an alternative name for the militant group, he added: “If our goal here is to destroy ISIL, we have got to do more than train a few folks in Syria and train a few folks in Iraq and drop some bombs. I just don’t know if that’s enough.”