Senate Majority Leader Harry Reid (D-Nev.) tamped down excitement over a $3.7 trillion deficit reduction plan by the Senate’s “gang of six” Tuesday by delivering a healthy dose of reality: Any bill dealing with revenues must begin in the House, and there’s likely not enough time to clear such a deal.

With less than two weeks left before the Treasury Department’s Aug. 2 deadline to raise the nation’s debt limit, Reid has been working with Minority Leader Mitch McConnell (R-Ky.) on a contingency agreement to avert default. Nothing about the Tuesday morning meeting with Senators and members of the bipartisan gang — nor the president’s praise of their work — changed the reality that time is running short.

“We only have 13 days — 13 days — and there’s a number of Senators who said they’ll do everything they can to stop the debt ceiling from being increased,” Reid told reporters after the Democrats’ weekly Conference luncheon. “It’s something that we have to look at closely — you know there’s no legislative language — and understand that before we start passing a bill out of the Senate, it has to start in the House because it’s a heavily revenued bill.”

Senate Majority Whip Dick Durbin, a member of the gang, has pronounced the plan “not ready for prime time” because it is not in legislative form and has not been scored by the Congressional Budget Office. The Illinois Democrat agreed with the Majority Leader that there simply isn’t enough time to get it done by Aug. 2.

Reid noted that in the process of negotiating with McConnell a package that could include more than $1 trillion in cuts, CBO Director Douglas Elmendorf warned his office that any package that significantly tackles entitlements would take weeks to score. Elmendorf suggested it would take two weeks to consider the budget effects of legislation that included billions of dollars in entitlement changes, according to Reid.

President Barack Obama’s generic support for the gang’s deal — which would reduce the deficit by about the same amount as the “grand bargain” the president had been negotiating with Speaker John Boehner (R-Ohio) — was symbolically important, as was the support from many Senate Republicans, including Conference Chairman Lamar Alexander (Tenn.) and Health, Education, Labor and Pensions ranking member Mike Enzi (Wyo.).

But Senate GOP support, or even the backing of the president — who McConnell said needed to be onboard months ago for a plan to be legitimate — does not necessarily mean the proposal will receive sufficient votes in the tea-party-influenced House.

Democratic leadership aides played up the outpouring of support from Republican Senators to contrast their backing of a stricter Cut, Cap and Balance bill set to hit the floor this week. Republicans say that supporting both measures is not mutually exclusive.

Meanwhile, Reid was careful to play down the president’s remarks earlier Tuesday.

“I think ’embracing’ the plan is a little stretch,” Reid said of Obama’s statement. “But he had a conversation with one of members of the gang of six. He’s very interested in what it does, but the president knows, because I’ve explained it to him many, many different times, the problems we have with time around here. “

Reid has said the Senate will not take a day off until the debt ceiling is raised, and the long view is that a sweeping deal could clear Congress after an initial plan to raise the debt limit is approved. The Reid-McConnell proposal includes a joint commission to review deficit reduction ideas, and it would tentatively be required to submit a plan by the end of this year. The gang of six framework — bolstered by Tuesday’s support — could give that provision added heft moving forward.

Those involved in the gang of six talks would ideally like to get some of their provisions included in the Reid-McConnell deal. Sen. Mark Warner (D-Va.), a member of the gang, will report to Reid within the next 24 hours about any provisions that the gang’s Democratic members believe could fit into the package, the Majority Leader said.