The two key lobbying groups for the savings and loan industry are at war over who the White House should pick for the Democratic seat on the Federal Home Loan Bank Board, the federal agency that regulates S&Ls.

The furor centers on George J. Benston, a professor of economics at the University of Rochester, whose free-market philosophy has made him front-runner as President Reagan's choice to fill a vacancy on the three-member bank board. The vacancy will open when board member Mary Grigsby steps down, which could be as early as Aug. 1.

The controversy over Benston highlights a growing split in the S&L industry, which faces record failures, between those who want more deregulation and those who want less.

Fighting Benston's nomination is the powerful U.S. League of Savings Institutions, whose members represent 96 percent of the $1.1 trillion in assets insured by the bank board. The league generally opposes any deregulation that blurs the distinction between banks, S&Ls and other financial service institutions.

Bank board Chairman Edwin J. Gray, who has been criticized for being too close to the league, also is said to oppose Benston as a nominee and supports the league's attempts to kill his nomination. Bank board sources say Gray has asked bank board employes to examine -- on government time -- Benston's published works and compile a list of objectionable positions he holds.

A bank board spokesman denied that employes had been used, saying there is no need for such scrutiny because the bank board is already familiar with his work.

In the opposite corner, favoring Benston's nomination, is the National Council of Savings Institutions, a three-year-old trade group whose members represent 22 percent of the assets insured by the bank board. The council's members, which tend to be larger thrift institutions, generally favor deregulation.

Although the council is smaller than the league, its power is growing, industry sources say. The council's president, for example, is John H. Rousselot, a former Republican congressman from California with close ties to the White House.

And although the league and the council share many members, the difference of opinion between the two lobbying groups is sharp.

The White House, caught in the middle, could present a nominee to the Senate, which must approve the choice, as early as next week, congressional banking sources said. The White House had no comment.

The White House could sidestep the argument by postponing naming a nominee for the Democratic seat on the bank board and instead filling the Republican slot that will open when board member Donald Hovde steps down Aug. 31, the congressional aides said.

As late as last week, however, the White House was asking for names of possible nominees for the Republican board members and may not have a choice ready before two of the bank board's three seats become vacant. Chairman Gray has said he will not step down before his term expires a year from now.

Because the bank board needs at least two members to pass regulations and take other actions, the White House is unlikely to leave the agency impotent at such a turbulent period in the S&L industry's history and so will press ahead to pick a Democratic nominee soon, S&L industry sources say.

But opposition to Benston, who specialized in studying the banking and S&L industry during his academic tenure, is fierce. The U.S. League recently sent letters to Sen. Jake Garn (R-Utah) and to the White House saying that the professor would be an "inappropriate" choice. In one letter, the league voices its opposition even though it "acknowledges the president's prerogative" to pick a nominee.

The U.S. League has said it would back Robert Mettlen, chairman of the Federal Home Loan Bank in Dallas, for the Democratic slot. But U.S. League Chairman Gerald J. Levy also has agreed to meet with Benston in the near future to discuss the state of the S&L industry, the league confirmed yesterday.

Hovde and Grigsby, at the request of the White House, are serving past the expiration of their terms while the search for bank board nominees continues. Hovde's term expired June 30, 1985; Grigsby's, last month.

Deregulation of financial services and a drop in the price of oil and real estate have shaken the thrift industry since 1980, causing a steady rise of record failures.

Benston's studies of the S&L industry's problems conclude that thrifts should be allowed to diversify outside home lending. Among his most controversial works are studies concluding that direct investments by S&Ls do not necessarily make the financial institutions more unstable and more likely to fail.

In other studies, including one for a bank trade group and another for the National Science Foundation, Benston concluded that in the neighborhoods he looked at, evidence was too skimpy to support allegations by community groups that the areas were being redlined.

"I did not say there has never been any redlining," he said, referring to the notion that banks avoid making loans to economically depressed areas such as inner cities, where residents often are predominately black or Hispanic. "No researcher could say that."

He noted he has also said many times that until the mid-1960s, the Federal Housing Administration routinely redlined certain areas.

The league is distributing quotes from Benston's published works that appear to show that Benston has said he favors complete deregulation and that he believes no neighborhood has ever been redlined.

"I don't believe in deregulation as a matter of faith," Benston said. "I'm willing to change my point of view, depending on the evidence.