The announcement Sunday came after Republicans, including the top two senators on the Banking Committee, warned they would not vote to move the legislation forward now because of differences with Democrats on key issues.
Senate negotiators said on televised Sunday news talk shows they're nearing a deal on financial reform but it won't come by Monday.
"We don't have a bipartisan compromise yet, but I think there's a good chance that we're going to get it," Senate Minority Leader Mitch McConnell, R-Ky., said on "Fox News Sunday."
Sen. Bob Corker, R-Tenn., a negotiator on the reform bill, said on ABC's "This Week" Republicans would withhold support until they reached agreement with Democrats on key issues.
"We don't need to address every issue in this compromise," he said, while stressing the need for oversight of derivatives, more consumer protection and "orderly liquidation" of failing banks.
"If we can get that template agreed to in a bipartisan way, then we can debate some of the amendments," including his, he said. "But I think it's very, very important that we reach that bipartisan agreement first because in the Senate, as you know, it takes 60 votes to change anything."
"If we keep working together, we'll get a bill," Sen. Richard Shelby, R-Ala., a member of the Banking Committee, said on NBC's "Meet the Press" Sunday. "It might be this week. It might be next week."
He and the committee's chairman, Christopher Dodd, D-Conn., who also appeared on the program, said they would continue working on the bill.
Many Republicans have complained Democratic reform proposals would allow future bailouts by the federal government. Democrats note their proposed legislation specifically prohibits bailouts.
President Barack Obama said Saturday that unless Congress approves the bill, the country will be at risk of another economic meltdown and further government bailouts.
Lawrence Summers, the director of the White House's National Economic Council, said on CBS's "Face the Nation" Democrats and Republicans seek many of the same elements of financial reform.
"If you look at the things the experts have cited as causes (of the economic downturn) ... they are all addressed by this bill," Summers said.
The Senate bill -- the House has already passed its version -- would overhaul Wall Street regulation and establish a new consumer protection agency overseeing and enforcing lending rules to prevent unfair practices by banks and credit card companies.
Treasury Secretary Timothy Geithner told CNN Sunday the financial regulatory system "didn't work so good for our country," leading to the recession.
"Our system as a whole is not going to just be more stable (after reform) but I think our institutions are going to be much stronger," he said.
"The basic strategy that guides reform -- and it's guided our response to the crisis -- is you bring more transparency, bring more disclosure, you force more capital into the system earlier, and you make sure you put in place basic things that make markets work better."