The one hanging issue on both the House and Senate tax bills is oil, and it is turning reformers into backroom negotiators, pitting a Republican oil-state senator against a Democratic Texas colleague and forcing a nervous administration to hedge on campaign criticism of the windfall profits tax.

In the Senate, where the tax bill has been on the floor for five days, action came to a near halt yesterday as Sen. Robert J. Dole (R-Kan.), no enemy of the industry, tried to figure out how to head off a pro-oil amendment sponsored by Sen. Lloyd Bentsen (D-Tex.).

While confident he has the votes to beat the Bentsen amendment, Dole would prefer to avoid a record vote on the issue -- exempting 1,000 barrels of oil a day from the oil profits tax -- which is close to motherhood in states where oil is part of the economic lifeblood.

"If it comes down to a choice between mothers and oil, there will be a lot of dead mothers," one House aide, delightedly watching the Senate stalemate, said yesterday.

An aide to Bentsen, however, said the senator will not budge and intends to force colleagues, including Republicans tied both to the oil industry and to the Reagan administration, to vote against the exemption as a "matter of principle."

At the same time, House Democrats on the Ways and Means Committee, in casting about for a key oil negotiator for their bill, have come up with the unexpected choice of Rep. Richard A. Gephardt, a reform Democrat from St. Louis.

Gephardt came to Congress in 1977 vowing to do all he could to eliminate the influence of special interests. For the past week, he has been meeting behind closed doors with congressmen from Texas, Louisiana and Okahoma, attempting to determine what oil sweeteners will acquire, in polite language, their votes. t

"We've taken to calling him J.R.," a southern congressman said, referring to the lead character in the television show "Dallas," "but I'm not sure he likes it." An aide to Gephardt said he is taking the kidding well: "He laughs with an oily grin."

At stake in the negotiations on both sides of the Capitol are, at a minimum, $1.6 billion in tax breaks for oil, mainly targeted to independent producers, and conceivably in excess of $4 billion if the Bentsen amendment and a catch-all of other proposals were to pass.

The basic starting point in all deal-making is generally conceded to be the present terms of the Senate Finance Committee bill on the floor for debate. It calls for the phased-down reduction of the profits tax on new oil -- oil discovered after Jan. 1, 1979 -- and a $2,500 annual tax credit for royalty holders.

Together, these two will cost the Treasury about $2.1 billion a year by 1986. Dole spent most of yesterday attempting to negotiate an arrangement with Bentsen under which the Texas senator would offer his thousand-barrel-a-day exemption, but then would permit some other oil-state senator or Dole to propose a less costly substitute amendment.

Dole said he is considering the elimination of the profits tax on new oil in a process to be phased in over four to six years. Most new oil is owned by independent producers and the accelerated tax break would cost an additional $1.6 billion in 1986.

Administration sources said they are watching this dueling over new breaks for oil with some trepidation because of the potential these conflicts have to force up the cost of the tax bill and to endanger the administration's commitment to balance the budget by 1984. Voicing these concerns, however, is difficult for administration lobbyists, because President Reagan has been highly critical of the profits tax in past campaign statements.

In the formal action in the Senate yesterday, an amendment giving users of the income tax short form the right to deduct a portion of charitable contributions was passed by a vote of 97 to 1, and privately owned companies were exempted from giving employes voting privileges when they provide workers stock under what are known as employe stock ownership plans.