The measure, which some senators earlier said could be released as early as Thursday, is expected to go before the Senate Judiciary Committee next week and advance to the floor of the Democratic-controlled Senate for a vote before the Memorial Day holiday weekend at the end of May, Sen. Robert Menendez, D-N.J., said.
Judiciary Committee Chairman Patrick Leahy, D-Vt., said the committee would have an immigration hearing Wednesday.
But hurdles remain, The Washington Post said, especially in the GOP-controlled House, where most Republicans say they oppose letting unauthorized immigrants become citizens.
Menendez, one of the bipartisan Gang of Eight senators working on the agreement, told an immigration-reform rally outside the U.S. Capitol Wednesday he believed the compromise Senate bill would become law by the end of the year.
"It is in the nation's interest, in the economic interests of the United States and in the security interests of the United States," said Menendez, who spoke in English and Spanish.
Many at the rally called for the law to offer a direct path to citizenship for the estimated 11 million immigrants living in the United States without legal permission.
The Gang of Eight has largely agreed on the bill, which several news organizations report would require tough border measures before people living in the country illegally can take the first steps toward becoming U.S. citizens.
As drafted, the legislation would provide up to $3.5 billion for the Department of Homeland Security to set up a five-year border security plan.
President Barack Obama, who has made immigration reform a second-term priority after winning 71 percent of the Latino vote in November, has vowed to introduce his own bill if the senators are unable to produce a proposal this month.
"I believe that we can get comprehensive immigration reform passed," Obama said at a Democratic National Committee fundraiser in California a week ago.
"That is going to mean that America can continue to be a nation of laws, but also a nation of immigrants, and attract the best and the brightest from all around the world," he said. "And if we push hard and we stay focused, we've got the opportunity to get this done over the next couple of months."
Lawmakers first said they hoped the proposal would be ready in March, then in early April.
Meanwhile, immigrant advocates say 1,000 people are being deported every day.
Obama has told the advocates he will still do the deportations because stopping them would anger the GOP when he needs its support for the comprehensive bill, the Post said.
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