The rebellion is being led by David Boren of Oklahoma, a Democrat who is a key member of the Senate Finance Committee. The Democrats' slender 11-9 majority in the committee means that his defection would be enough effectively to kill it.
Earlier this week Mr Boren, whose state is one of the country's main energy producers, declared he would oppose the tax, designed to raise dollars 71bn, 'in committee, on the floor, anywhere'. Three or four other Democratic senators largely share his view. Further defections could even cost the Democrats their Senate majority of 57-43, in any case likely to be reduced by one if the Republicans make their expected capture on 5 June of the Texas seat formerly held by the Treasury Secretary, Lloyd Bentsen.
Faced with a challenge that could destroy Mr Clinton's credibility in Congress, the White House is mounting a counter-offensive, arguing that the Boren plan, of deep cuts in welfare and other programmes, would merely penalise the poor and the elderly. 'Strip away the rhetoric,' said the President on Thursday, 'and there's dollars 40bn of burden on people just above the poverty line and dollars 40bn less on those of us who can afford to do a little more for our country.'
The immediate worry for the administration is that the very prospect of fresh trouble in the Senate could destroy the fragile majority for the Clinton programme in the House of Representatives, which is due to vote on it next week. Although the Democrats have a House majority of 80, misgivings about the energy tax are widespread. Many Congressmen are deeply unhappy about supporting a measure unpopular in their own constituencies, and which might now come to grief in the Senate.
But in wider terms, the energy tax dispute symbolises the difficulties of the Clinton presidency. The resistance by moderate Democrats, such as Mr Boren, reflects a growing perception - reinforced by attacks from the Republicans and a resurgent Ross Perot - that Mr Clinton is not the 'New Democrat' he portrayed himself as throughout the election campaign, but merely an old school 'tax-and-spend' liberal.
More than once during his trip this week to New Mexico and California, the President found himself under fire for reneging on his earlier promises of tax cuts for the middle class. Opinion polls, too, confirm that, while the public is prepared for sacrifice to reduce the deficit, it far prefers that this be done by cutting expenditure rather than by increasing taxes.
However, rounding out the dilemma for the President is the knowledge that to give way on the energy tax will cement an impression of weakness: that at the first whiff of opposition he instinctively reaches for compromise. Last month's demise of his dollars 16bn stimulus package heavily dented his authority on Capitol Hill. Surrender now would cost him further precious capital. In such a climate, the prospects of the health-care reforms due to be unveiled next month would be dismal indeed.