But Sen. Lindsey Graham, R-S.C., another member of the Gang of Eight senators of both major parties in talks to develop a broad immigration-reform proposal, said it would probably be a couple more weeks before the plan is completed.
"All of us have said that there will be no agreement until the eight of us agree to a big, specific bill, but hopefully we can get that done by the end of the week," Schumer said Sunday on CBS News' "Face the Nation."
When asked to confirm the end-of-the-week time frame, he said, "Hopefully ... that's what we're on track to do."
But Graham told NBC's "Meet the Press," "We're hoping to get this thing done in the next couple of weeks."
He said a hangup was a guest-worker agreement between the nation's top business and labor groups for low-skilled immigrants worked out the end of March. That deal was widely touted as clearing the path for the broad immigration legislation to be introduced this month.
But Graham said the Senate group was now "revisiting" that agreement, between the AFL-CIO and the U.S. Chamber of Commerce, which the construction and some other industries complained about.
"How can you access it in an affordable fashion when you can't find an American worker?" Graham said.
"If we're reasonable with 11 million, if we all give them a pathway to citizenship that's earned and hard and fair, get in the back of the line, pay taxes, learn the English language, then the Democratic Party has to give us the guest-worker program to help our economy. That's what we're arguing over."
Graham issued a statement a short time after the program ended saying he was confident the agreement would hold.
Sen. John McCain, R-Ariz., who is also part of the Gang of Eight and appeared on "Face the Nation" alongside Schumer, predicted a tough road ahead for the legislation.
"There will be a great deal of unhappiness about this proposal because everybody didn't get what they wanted," he said, with Schumer agreeing. "There are entrenched positions on both sides of this issue, as far as business and labor."
Sen. Marco Rubio, R-Fla., the son of Cuban immigrants and a member of the group, did not speak Sunday but said in a statement last week suggestions of an agreement were "premature."
"In order to succeed, this process cannot be rushed or done in secret," his statement said.
McCain told "Face the Nation": "I reject this notion that something is being railroaded through. This is the beginning of the process, not the end of it."
"The Judiciary Committee will act. There will be amendments. There will be debate. Then it will go to the floor of the Senate. There will be plenty of time for discussion and debate," he said.