Updated

Several prominent Republican senators accused their GOP colleagues of "secretly" negotiating a sweeping immigration overhaul, urging them to open up about the details by next week.

The letter from Republicans on the Senate Judiciary Committee marked the latest flare-up in the GOP caucus over the immigration plan, which has not yet been introduced. Republican negotiators have urged their colleagues not to pre-judge the proposal, but others are getting frustrated by the lack of hard details and concern that majority Democrats will try to "rush" the bill through after it's introduced.

"We believe it is critical that the public and the entire Senate body be given adequate time to read and analyze the contents of any immigration bill put forth by the majority," the senators wrote in the letter Thursday to the four Republican negotiators: Florida Sen. Marco Rubio, Arizona Sen. John McCain, Arizona Sen. Jeff Flake, and South Carolina Sen. Lindsey Graham.

"Because the president has failed to lead on this matter, your group has secretly met for months and not consulted with members of the Committee about major changes to our nation's immigration laws. The time for transparency has come," they wrote.

The lawmakers asked to be briefed by staff no later than close of business Monday, and asked for a caucus-wide briefing early next week "so that all members can raise concerns and questions before the deal is finalized."

In response, Rubio, R-Fla., said he agreed on the need for hearings and planned to brief all Senate Republicans next week. He disputed the claim that the immigration bill process has been more secretive than any other and said he's worked to incorporate suggestions from all senators.

Rubio has tried to act as a bridge between the bipartisan "Gang of Eight" and other congressional Republicans. His involvement is seen as key to bringing any immigration package across the finish line.

Rubio has recently voiced concern -- echoing the complaints of his colleagues -- that the process was being rushed.

Senate Judiciary Committee Chairman Patrick Leahy, D-Vt., in response to those concerns, said he would consider holding one hearing.

The Senate Judiciary Republicans who wrote the latest letter complained about this approach, predicting Leahy would "likely proceed directly to a mark-up of a far-reaching and complex immigration bill based on your group's proposals."

The letter was signed by top committee Republican Chuck Grassley of Iowa, Alabama Sen. Jeff Sessions, Utah Sen. Mike Lee and Texas Sen. Ted Cruz.

Rubio released a letter back that stressed his agreement on an open process.

"This proposal will be a starting point," Rubio wrote. "As members of the committee of jurisdiction I expect you to have ample opportunity to review, comment and amend as you see fit ... You can expect that I will continue to defend the rights of every senator, myself included, to conduct this process in an open and detailed manner."

The Associated Press contributed to this report.