Senate Majority Leader Harry Reid (D-Nev.) and Senate Banking, Housing and Urban Affairs Chairman Chris Dodd (D-Conn.) said they still believe they could pass a $700 billion economic bailout by Monday, even as they implored reluctant House Republicans to join their efforts.

“We could have this [written] by tonight, and vote on Sunday or Monday,” Reid said of negotiations among Senate Democrats, Senate Republicans and House Democrats. “We can’t do it alone. We need the House Republicans to join us.”

Reid also expressed frustration that the White House and Treasury Secretary Henry Paulson appear to be pushing back on some details of the tentative deal reached by a bipartisan group Thursday.

“There’s some degree of amazement from the White House … that people who are elected have constituencies that they have to take care of. … We don’t serve only Wall Street,” Reid said. He added that Paulson and Federal Reserve Chairman Ben Bernanke “are going to have to become more realistic” about what Congress can do.

Dodd said Democrats would insist on several principles, even as they sought a compromise.

“There are two or three areas that are non-negotiable,’ Dodd said. “The language of them is negotiable but not the principle.”

Among those non-negotiable items is a demand that executive compensation be limited, that taxpayers reap some benefits if the markets rebound, and that there be strong oversight of the program to buy up bad mortgages from struggling financial institutions.

Reid and Dodd said they were heartened that House GOP leaders were sending Minority Whip Roy Blunt (R-Mo.) to be their representative at the negotiating table today. Reid complained that the presence of House Financial Services ranking member Spencer Bachus (R-Ala.) at yesterday’s bipartisan meeting was pointless because he “had no authority to do anything.”

Reid announced that staffers were working furiously to draft changes proposed at last night’s meeting with Paulson, and Members hope to meet later today to discuss those changes.