A key Republican lawmaker said Sunday that a bipartisan group of senators does not yet have an agreement on a proposal to overhaul the nation's immigration laws, despite making progress in recent days.

Marco Rubio

Sen. Marco Rubio (R-Fla.), a member of the eight-person group, said in a statement  that he is encouraged by reports that labor and business leaders have reached a deal over a new visa program for low-skilled foreign workers. That program was considered crucial for the senators to move forward on the comprehensive legislation, which also will feature a path to citizenship for the nation's 11 million illegal immigrants.

But Rubio emphasized, "reports that the bipartisan group of eight senators have agreed on a legislative proposal are premature."

He said the group has "made substantial progress" and he expected the group to eventually reach agreement. But he added that the legislation will require "a healthy public debate," including committee hearings and input from other senators to offer amendments.

Rubio, whose parents emigrated from Cuba, is considered a key GOP member of the group and a potential presidential candidate in 2016 who has support in the Hispanic community at a time when Republicans are attempting to broaden their appeal to minority voters.

But he is also a tea party star who has supported immigration reform while treading gently through the process of developing a bill. This weekend, he sent a letter to Sen. Patrick Leahy (D-Vt.), chairman of the Judiciary Committee, which will oversee markup of an immigration bill, asking him not to rush the process.

In his statement Sunday, Rubio said: "Arriving at a final product will require it to be properly submitted for the American people's consideration, through the other 92 senators from 43 states that weren't part of this initial drafting process. In order to succeed, this process cannot be rushed or done in secret."