With a small group of Senators reportedly on the verge of striking a border security deal on immigration reform, Senate Majority Leader Harry Reid said he may start the parliamentary process on Thursday to force an end to debate in order to bring about final action next week on sweeping immigration legislation.

"One of my goals is to work really hard to try to finish as much of this bill as soon as we can," Reid said on the Senate floor Wednesday evening, as he threatened to file cloture - a motion to shut off debate - as soon as Thursday.

"I have told everybody many times - we are going to finish this immigration bill," Reid added.

Reid's comments came as a bipartisan group of Senators continued to work on a deal that might win over reluctant Republican Senators, who keep saying they want tougher border measures included in the bill.

CNN reported that plan could be unveiled as early as Thursday morning.

GOP Senators demanding extra border security provisions seized on part of a Congressional Budget Office report that was issued on Tuesday, which estimated that "under the bill, the net annual flow of unauthorized residents would decrease by about 25 percent relative to what would occur under current law."

In other words - illegal immigration into the United States would slow only by about 25 percent, as compared to doing nothing - so, GOP Senators say that shows more has to be done on border security.

The plan being worked on behind the scenes by Senators in both parties would add as many as 20,000 agents to the Border Patrol as well as extra fencing and technology along the Mexican border.

"Our whole effort has been to build a bipartisan group that will support the bill," Sen. John Hoeven (R-ND), one of the Republicans trying to negotiate a deal, told the AP.

It wasn't clear how many Republicans it might bring over to the legislation, even as GOP Senators urged change.

"The Gang of 8 bill is 1986 all over again," said Sen. Ted Cruz (R-TX), referring to the last major immigration law that gave citizenship to millions of people in the U.S. illegally.

"Amnesty with empty promises to secure the border," Cruz added.

Whether a deal on border security can be struck or not, Democrats were looking at the calendar, figuring out when to file a cloture motion to force an end to debate.

If Sen. Reid wants a final vote by next Thursday or Friday, then he may well have to file cloture by this Thursday or Friday, because of the time that is needed to get past those procedural hoops and hurdles.