To the mortification of Democratic leaders, the pivotal role in passing President Reagan's budget and tax policy may be played by a former Texas A&M economics professor, serving his second term in Congress, named Phil Gramm.

Rep. Gramm, representing a rural district in east central Texas, is a leader of 44 newly militant conservative House Democrats. Behind closed doors, he has been negotiating with budget director David Stockman and Rep. Jack Kemp about bipartisan conservative budget and tax policies. That points to possible humiliation for speaker Thomas P. O'Neill and friends in the Democratic-controlled House.

That would be sweet revenge for conservative southern Democrats such as Gramm, the "boll weevils" who have been treated by their liberal colleagues with increasingly less respect the last two decades. Suddenly, the good ol' boys who sit together on "redneck row" on the House floor are pressuring the Democratic leadership to satisfy their wishes or face a reborn Dixie-GOP coalition that could carry the day in the House.

Contempt for the boll weevils by the party's dominant liberals remains undiminished. Breakfasting with newsmen April 2, Sen. Gary Hart of Colorado bemoaned the Reagan administration's refusal to compromise on its social welfare cuts in order to create a bipartisan budget. When pointed out to him that some Democratic senators were consistently supporting the Reagan cuts without benefit of compromise, Hart snapped that they were "out of the mainstream of the party."

Mainstream or not, a hard-core dozen Democratic senators opposed efforts (consistently supported by Hart) to beat back the Reagan budget cuts. Although their votes are not as crucial in the Republican-controlled Senate, they did guarantee a comfortable victory for Reagan on every budget question. Without 15 Democratic opponents, liberal Republican Sen. John Chafee's effort to restore $1 billion would have carried by one vote.

Gramm, a member of the House Budget Committee, wants to repeat this in the House. He and his fellow boll weevils will break Democratic ranks unless the majority Democratic budget position accepts two non-negotiable demands: First, it cannot authorize a penny more than Reagan's own budget; second, it cannot cut a penny out of defense.

This poses a dilemma for the budget committee chairman, Rep. James Jones of Oklahoma. A moderate himself, he will lose platoons of Democrats if he meets Gramm's demands. But if he does not, he risks being rolled. Gramm, who publicly declares himself closer ideologically to Dave Stockman than to Tip O'Neill, met with Stockman Monday night to negotiate a bipartisan budget for the House.

The prospect of a similar Dixie-GOP coalition on the tax question is more complicated. Gramm has publicly declared the intellectual triumph of Jack Kemp's supply-side economics, but in the next breath opposes the cut in individual tax rates that is the heart of it. Ex-professor Gramm, while extolling supply side doctrine talks in Keynesian idiom about "stimulation" of the economy and measures the tax cut in static revenue loss.

Gramm has been conferring with Sen. David Boren of Oklahoma, a key member of the nameless organization of conservative Democrats similar to the 44-member Conservative Democratic Forum in the House. Gramm and Boren, making an invalid distinction between business and individual taxes, want to reduce Reagan's individual tax cuts (which they fail to understand would provide the only incentive for countless small business).

Yet, tax differences between the conservative Democrats and Reagan should not be overstated. Boren and the influential Sen. Sam Nunn of Georgia do not rule out all individual rate reductions. Gramm backs an across-the-board 10 percent cut for the second and third years of Reagan-Kemp-Roth, but not the first. Neither the Senate nor the House boll weevils want further income redistribution against the rich in the guise of tax reform.

The three Conservative Democratic Forum members on the House Ways and Means Committee -- Reps. Ken Holland of South Carolina, Ed Jenkins of Georgia and Kent Hance of Texas -- may not like Kemp-Roth, but they like liberal social engineering even less.

Thus, unless Ways and Means tailors its bill to their measurement, the boll weevils might well vote for the Republican substitute. That would be remotely possible even if the substitute were straight Reagan-Kemp-Roth, but more likely if Kemp and Gramm are able to hammer out something in their current conversations.

All this is heady stuff for Phill Gramm, who just a year ago as a freshman congressman was snubbed by party leaders for trafficking on economic issues with the likes of Michigan's Rep. Dave Stockman. Now, he and his fellow boll weevils feel closer than their congressional leaders to the consensus among ordinary Democrats. Informed of Gary Hart's remarks, Sen. Boren told us: "We may be closer to the mainstream than he imagines."