What do you do when you hold the government franchise to build the most expensive private construction project ever conceived, but can't find a banker who will touch the project?

If you're a politically connected pipeline tycoon by the name of John McMillian, you put together a lobbying team that covers the waterfront from old friends like Walter F. Mondale and Robert S. Strauss to new ones like former Reagan speechwriter Peter Hannaford.

You commission sheaves of reports on how your project, the Alaska natural gas pipeline, serves the national interest. You arrange for help in the persuasion game from the unions that will build the line, the steel companies that will supply the raw materials and the oil companies that will feed the gas through it.

Then you let the White House and Congress know that if they want the private sector to finance the $40 billion project, they're going to have to shift some of the risk of building it from the bankers to the consumers.

McMillian, a hard-driving, one-time, oil field roustabout, is in the midst of such a campaign. His company, Northwest Alaskan Pipeline, is seeking a series of waivers to its 4-year-old franchise for the project, the most controversial of which would expose nearly half the nation's natural gas customers to the possibility of being billed for the line before it is finished.

The pipeline is expected to provide the "lower 48" states with more than 4 percent of their natural gas needs by the time it is completed in 1986 at the earliest. It is planned to run a total of 4,790 miles--six times longer than the trans-Alaskan oil pipeline--from Prudhoe Bay, Alaska, to the Midwest in one segment and to California in another.

Consumer groups are opposing the waiver package. The Reagan administration, after some initial hesitance and substantial tinkering, is supporting it. But so far its efforts have been modest, and the major lobbying thrust is being orchestrated by McMillian, who is drawing on a lifetime of political friendships to help steer it through Congress.

McMillian is a heavy financial contributor to the Democratic Party, and his aggressive partisanship has become a matter of regular comment on Capitol Hill, where the package is facing a key vote today in the House Energy and Commerce Committee.

Is it any coincidence, some wonder, that Rep. Tony Coelho (D-Calif.), chairman of the Democratic Congressional Campaign Committee, is one of the most active supporters of the package, and that this summer McMillian purchased a $10,000 table at a dinner sponsored by the committee?

Committee director Martin Franks denies any link. He did, however, note that McMillian has been "an old friend of the committee" and that "we are in the midst of a broad effort to make Democrats aware that you don't remain in the majority by screwing your friends."

Still, he said that the "widespread perception" that campaign contributors receive special treatment from Congress is "vastly overblown."

Former Democratic National Committee chairman Strauss, whose law firm helped McMillian draft the waiver package, took the same tack. "He's my friend and I'm helping him with pride," Strauss said. "But I'm doing it because it's a good project. I certainly don't go around getting involved with helping friends to the tune of $40 billion." Then Strauss added puckishly, "A couple of billion, maybe."

Some committee staffers say that if McMillian is getting any boost from his Democratic Party connection, it is costing him on the other side of the aisle. "He's a dominant personality and people tend to associate the pipeline project with him," said one Republican Senate staffer who asked not to be identified. "I've heard people say why should we rush to his rescue when the rotten bastard is a Democrat."

For his part, McMillian is inclined to dismiss the importance of partisanship. He notes that he himself has given money to individual Republican candidates, and that most of the nine other pipeline companies and three oil companies in his consortium tend to be more Republican in orientation.

Still, he is sensitive enough to the whole question of party loyalty to have assembled a lobbying team that is scrupulously balanced among Democrats and Republicans. To work the GOP side of the aisle, he retained the Hannaford firm and Jack Ferguson & Associates.

For Democratic persuasion, he is leaning on Strauss's firm, Akin, Gump, Strauss, Hauer & Feld, and the firm of White, Fine & Verville.

He also has hired Mondale as a consultant, although a spokesman for the former vice president said he has made only one call to Capitol Hill on the project. McMillian, who this spring contributed $5,000 to Mondale's fund-raising vehicle, the Committee for the Future of America, said the main reason he brought Mondale on board was to help in arranging for financing on the world market.

When McMillian won the franchise in 1977 over two competing pipeline consortiums, he assured Congress he could build the pipeline without government loan guarantees and without consumer risk-sharing.

He now says high interest rates and inflation have thrown his projections out of kilter. To placate a now-skittish financial community, the waiver proposal would split the pipline project into three segments and call for the Federal Energy Regulatory Commission to set a "date certain" (probably around January, 1987) by which all three are expected to be finished. If, on that date, any one segment is completed but the others are not, the pipeline consortium could begin billing customers for the completed section, even though they would not be receiving Alaskan natural gas.

"This is the minimum the banks can live with, and we're not even sure it is enough," McMillian says. "But we do know if we don't get it, we have no chance for private financing."

The chanciness of the project, even with the waiver package, is part of what is slowing its progress through Congress. Some fear that the waiver proposal is just a foot in the door, that McMillian eventually will be back asking for full federal loan guarantees.

Meantime, the likelihood that natural gas prices will be decontrolled by the time the line is completed has caused some to wonder whether the Alaska gas, with its high fixed transportation costs, will be economically competitive with other forms of energy.

Complicating all this, from McMillian's point of view, is the lukewarm response from the Reagan administration. It substantially cut back on his original waiver request (which had been for a much more extensive pre-billing system) before approving it, and so far it has done little lobbying for the package on the Hill.

But he hopes the administration will get more involved, in part because the proven gas reserves in Alaska are 13 percent of the nation's supply, and in part because letting the project die would severely strain relations with Canada, which has a heavy stake in it.

"We need the best lobbyist in town in on this one," said McMillian, with a Democratic tipping of the hat to President Reagan. "We don't have him yet, but we're hoping."