John Sununu’s office has just released a pretty clear answer on cloture for the immigration bill:

RE: Future cloture vote in immigration. On this bill, no. On this bill with any amendments proposed by bipartisan group, no.
Senator Sununu opposes the immigration bill – voted against it last year, and has voted twice against cloture on immigration this year – both on June 7.

Another like-minded Republican senator’s office noted that bill opponents have three bites at the apple, so to speak, not counting any “poison pill” amendments that could make the deal unworkable. The first is the vote to bring it back to the floor; the second is the cloture vote to cut off further debate, and the third is on the bill itself.