Typically, House rules require that legislation be available for 72 hours before a vote. In order to bypass that, they'll have to have two separate votes to suspend the ordinary rules process. Presumably, if a deal is struck, this won't be a problem. After the rules suspension, the House will hold a vote. Again, presumably if a deal gets struck, it will pass and go to the Senate. In the Senate, they'll need to get unanimous consent (UC) to bypass the official Senate calendar and bring it straight to the Senate floor to get a vote. If that goes through, it's off to the President for signing.
The alternative way forward for a budget is for Harry Reid to move first to bring the budget passed yesterday by the House to the floor of the Senate with an amendment for whatever deal gets struck. This also requires unanimous consent. They'd pass the budget, send it back to the House for a vote on the amendment.
The key in this procss is unanimous consent. If a single Senator objects (leading candidates for something like that would be ideological purists like Sen. Bernie Sanders or Sen. Rand Paul) it's a likely multi-day process for a bill to go through. Obviously, that would not prevent a shutdown.
That's a quick overview of the dance that party leaders have to go through right now.