"The American people have a right to know what happened," Boehner said on CBS' Face The Nation.
The lawsuit would be spearheaded by Rep. Darrell Issa, R-Cal., who chairs the House Committee on Oversight and Government Reform.
Attorney General Eric Holder, held in civil and criminal contempt of Congress on a House vote last week, has said his department has turned over thousands documents, and that Issa and some Republicans are playing politics with the Fast and Furious.
Because of the contempt vote, "an unnecessary court conflict will ensue," Holder said recently. "My efforts to resolve this matter short of such a battle were rebuffed by Congressman Issa and his supporters."
At Holder's request, President Obama asserted executive privilege on the House's most recent document requests, setting up a potential legal battle.
The House, in a vote largely on party lines, held Holder in both criminal and civil contempt.
The Justice Department said Friday it would not prosecute Holder on the criminal contempt citation.
The civil contempt vote clears the way for a House lawsuit against the Obama administration.
