Senate Majority Leader Harry Reid moved Monday to end the debate on the immigration bill, filing a motion to set up final filibuster votes for Wednesday.

As he opened the session Monday afternoon, Mr. Reid made the parliamentary move to schedule the cloture vote — moving to keep to his schedule of trying to have the immigration bill done before he sends the chamber home for a July 4 vacation.

Some Republicans have already objected, saying that Mr. Reid is shutting off the debate too early.

As of Monday morning senators had introduced 372 amendments to the bill, but only a dozen of those amendments have seen votes, and there is little prospect that Mr. Reid will allow many more votes now that he’s moved ahead with the filibuster showdown.

“This is deeply, deeply disturbing. It is effectively shutting down the American people’s ability to be heard on this issue through their elected representatives,” the 13 senators said in a letter to Mr. Reid on Monday.

But Mr. Reid’s office said there’s been enough debate, and said the 13 senators who signed the letter are only trying to obstruct the legislation, not to improve it.

Reid spokesman Adam Jentleson also pointed to the committee action that preceded this month’s floor debate, and said in many cases it is Republicans who are blocking votes on amendments.

“This is the third week of floor consideration, and Senator Reid has been clear throughout that he wants to consider as many amendments as possible,” Mr. Jentleson said.

Senators will vote Monday night on whether to filibuster a new 1,200-page amendment to the immigration bill, which 11 Republicans and a handful of Democrats wrote and delivered to colleagues last Friday.

Opponents say they’ve had little time to go through the changes in the amendment.

But backers, including the Republicans who wrote it, say it only changes 119 pages of the 1,100-page bill.

“I’ve seen reports of a ‘1,200 page bill’ no one has read or had time to read. To be clear, the tough border and interior enforcement provisions that Sen. [John] Hoeven and I offered on Friday make up 119 pages added to the 1,100 pages that have been public since May,” said Sen. Bob Corker, Tennessee Republican.

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