WASHINGTON (CNN) -- Senate Majority Leader Harry Reid agreed Tuesday to delay bringing the immigration issue to the Senate floor until Monday in order to give bipartisan negotiations being conducted behind closed doors a few more days to finalize a deal, several Republican and Democratic aides said.

As late as Tuesday afternoon, Reid told reporters he was sticking with plans to bring immigration to the Senate floor Wednesday.

The group met Tuesday afternoon in what Sen. Trent Lott, R-Mississippi, called a "make or break" session to see if they could reach an agreement. Administration officials, Republicans and Democratic senators all emerged from that meeting reporting progress in their intense and highly complicated negotiations for comprehensive immigration reform.

An aide to Sen. Ted Kennedy of Massachusetts, a lead Democratic negotiator, said he asked Reid for more time because "the bipartisan negotiations are showing progress and this gives them a chance to continue."

GOP sources said they believe they are close to finalizing a deal, but said senators on both sides of the aisle have been demanding time to read the fine print of whatever negotiators agree to. Assuming a deal is struck over the next few days, moving the vote to Monday would give them that time.

Earlier, negotiators declined to discuss the details of their differences but several said they can't get a deal on the overall bill until each of the underlying issues is resolved.