"As a nation of immigrants, we must remember that we're all descended from people who came here from somewhere else in search of a better life," she said.
"But the broken immigration system we have now is unworthy of a great nation," she added. "It's time for Washington to tackle this problem head on."
Despite support from the White House, the AFL-CIO labor unions and the pro-business Chamber of Commerce, the bill's passage is by no means assured. Sixty votes are usually required to end Senate debate and consider adoption. There are currently 54 senators, including two independents, in the Democratic caucus, and 45 Republicans.
Leaders in the Democratic-led Senate want a final vote on the legislation by July 4.
The Republican-led House, meanwhile, is taking a smaller, piecemeal approach to the issue. Many of the components of the Senate bill are likely to find strong opposition there, giving House Republicans greater sway even before the Senate votes.
"What they have in the Senate has zero chance of passing in the House," said Sen. Rand Paul, R-Ky. "So, why not come to a conservative like myself and say, he's willing to work with you, why not work with me to make the bill closer to what would be acceptable in the House?"
Sen. Ron Johnson, R-Wis., said he remained hopeful a bill could be passed but said the bill would have to see changes if it stood any chance in the House.
"It doesn't do anybody any good just to pass in the Senate," Johnson said.
Immigration also has deep political implications.
In 2012, President Barack Obama won re-election with the backing of 71 percent of Hispanic voters and 73 percent of Asian voters. A thwarted immigration overhaul could send those voting blocs more solidly to Democrats' side in future elections. That has led some Republican lawmakers to support immigration reform, but the party's conservative base still opposes any legislation that would create a pathway to citizenship for immigrants living here illegally.
Paul and Johnson were on "Fox News Sunday."