Unless a bill specifically raises taxes or appropriates spending, it does not have to originate in the House. Even the bill being used for the Boehner plan originated in the Senate - it is Senate Bill 627, amended to include the language of the Boehner Plan using a technoque often called “gut and amend”.
Even if he wanted to include taxes and new spending, Reid could use the “gut and amend” process to insert his bill into a bill that “originated” in the House.
Reid: “Laws...we don’t need to stinkin’ laws.”