As a matter of principle it can be amended. But Frist cut off a good deal of the amendment range with his introduction of a cloure motion, as well as calling for a DIFFERENT measure (lobbying reform) to be debated before the cloture vote on the Border Control bill.
As a practical matter, there is so much contention about the details of a suitable guest worker program, that bringing discussion of that to the Senate flor won't happen. The Judiciary Committee has itsel;f pretty well knotted up on that discussion.
Any chance of the Judiciary Committee dropping this bill, even though it seems they all agreed on it now? I wonder how much effect being flooded with calls will help. Plus the House heros warning them that it won't fly in the House.