Both said they think the Senate should pass an extension of unemployment benefits, COBRA and the $8,000 tax credit for first time homebuyers. Schumer said the Senate would act this week on unemployment benefits.
"We're going to pass a bill this week, put it on the floor," Schumer said. "I believe it will pass. It will extend unemployment benefits for four weeks for all states and another 12 or 13 weeks for all states above 8 percent [unemployment]."
The House has already passed a bill that would extend benefits to Americans currently unemployed.
But Schumer said the Senate version would "go beyond what the House was. We think that, you know, 8 -- you have 8.5 percent unemployment, that's pretty bad, and you ought to get some help."
Both Senators also agreed that "throwing money at the problem" is not the answer, though.
"I think there are things we need to do to help people who need help, like unemployment benefits and the like," Cornyn said. "But I think throwing more money at the problem and racking up more and more debt for children and grandchildren is not the answer."
During his address Saturday, President Barack Obama acknowledged his concern over the climbing jobless rate and said he's exploring "additional options to promote job creation."
The president's remarks have raised the question whether the administration is rethinking about recovery efforts, possibly laying the groundwork for a second stimulus package.
On "This Week" today, Greenspan said he does not think there's a need for a second stimulus and warned that doing too much would be counterproductive.
"I think the focus has got to be on trying to get the economy going," he said. "But also have to be careful in trying to do too much would be counterproductive."
With 60 percent of the existing stimulus funding still to come, Greenspan said he is taking a "wait and see" approach.
In the second portion of the show, Schumer and Cornyn wouldn't commit to supporting a broader second stimulus. Cornyn said the stimulus isn't working and used Friday's record-high job loss report as proof.
"Well, I think the stimulus so far has been unsuccessful in achieving the goals the president set out for it," Cornyn said. "[Obama] said, with the stimulus, we'd seen 8 percent -- no higher than 8 percent unemployment. And we now see, with 60 percent of the stimulus unspent, that that has not been successful. We're going to see unemployment over 10 percent."
Cornyn said the American people are "scared" that spending and the debt is being "racked up."
"Forty-three cents out of every dollar spent in Washington today is borrowed," Cornyn said. "And the American people are justifiably concerned, even scared, that the spending and the debt that's being racked up. And unfortunately, the health care issue threatens to make matters worse, not -- not better."
Schumer said the current stimulus is working but stopped short of endorsing another one, saying the full effect of the government effort is not yet clear.
"So before doing a second stimulus, let's see how the rest of the 60 percent works and try to deal with the pain of some people in terms of the job front, where John and I agree, and in certain targeted areas of the economy, such as housing," he said.