The house actually has the primary power of the purse as per the constitution all spending bills must originate in the house. The senate can amend such bills (up to and including removing everything except the title) and then send that back to the house to vote on, but spending bills must originate in the house, and any adjustments to a house-proposed bill by the senate must go back to the house for a vote.

I only bring this up to be clear that the senate may reach a compromise bill, but the speaker would have to bring said compromise bill to vote on the floor and it would have to be voted on before it becomes law. The Senate is only the final stop if the approves the original 7 week CR that was passed by the House at the end of September which seems unlikely at this point. Negotiations with the house could very well extend the duration of the shutdown, even if the Senate reaches an agreement in principle.