Gutiérrez 's comments came on the same day that Sen. Rand Paul (R-Ky.), a rising star in the GOP and a potential presidential candidate, unveiled his own immigration plan.
Paul appeared to endorse a pathway to citizenship for undocumented immigrants during a speech in Washington, but later said that he does not support a "special" path to citizenship. Still, Gutiérrez praised Paul for putting forth a plan that includes legalization and said that it's a sign that the Senate has enough votes to pass a bill.
"They started with eight -- four Republicans and four Democrats in the Senate -- but you're going to need 60 votes," he said. "I think they're there on the 60 votes. And if they're there on the 60 votes, you're going to get a vote on immigration reform in the Senate."
The congressman admitted that obstacles remain, including a race against the clock. He said he is hopeful that the House and Senate will put forth their formal plans in April after Congress' Easter recess.
"We are under a time pressure, not one set by the president, Senate or the Speaker of the House or Democratic leader [Nancy] Pelosi," he said. "We are under a time pressure to resolve this issue because the moment is politically right. And I believe the further we get away from the day of Election Day, Nov. 6 of 2012, the less urgency there will be and the less likelihood of success."
Gutiérrez added that it's critical to resolve the issue quickly, possibly as soon as July, to stem the tide of deportations of undocumented immigrants, particularly those who have families in the U.S. and have not committed crimes aside from illegal entry.
The congressman said that the future flow of low-skilled immigrant workers, under a guest-worker program or another type of program, also remains a sticking point between Republicans and Democrats in Congress. But he said that the tenor of negotiations with Republicans have indicated to him that immigration reform will succeed.
"I think it's important to underscore that all the talk about a new attitude and a new approach to immigration by Republicans is true," he said. "I have seen it in every conversation with Republicans I have had."
ABC's John Parkinson contributed to this report.