Japan has mounted an unusually aggressive lobbying and public relations campaign to blunt the congressional reaction to the sale of computerized milling machines that have allowed the Soviet Union to make submarine propellers quiet enough to avoid U.S. detection.

The Japanese have enlisted senior officials of the Reagan administration in their attempts to soften legislative proposals, and are engineering a grass-roots campaign that started with letters to the 11-member Tennessee congressional delegation from Gov. Ned McWherter warning that sanctions could cost their state jobs.

"I think it is one of the biggest lobbying-public relations campaigns ever launched," said Clyde Prestowitz, a former Reagan administration trade negotiator who is finishing a book on U.S.-Japanese trade relations.

It is unclear how much the Japanese are spending on the effort, since their U.S. representatives have not yet filed full statements, including their fees, with the Justice Department. But the outlays would be in addition to the estimated $60 million a year that Japanese companies and their government spend to influence public opinion and legislation in the United States.

Justice Department officials estimate that wholly owned American subsidiaries of Japanese corporations spend another $25 million to $30 million on lobbying and public relations efforts that do not have to be reported to the government.

The new campaign has been mounted by Toshiba Corp. -- the electronics conglomerate whose subsidiary joined with an arms maker owned by the Norwegian government to sell the sophisticated equipment to Moscow -- and two other Japanese giants, C. Itoh & Co., the large trading company that played a role in the sale, and NEC Corp., an electronics firm that has been named in Japanese press reports as a principal in other illegal sales of sensitive technology to the Soviets.

The companies intensified their lobbying and public relations efforts after the Senate overwhelmingly approved legislation that would bar U.S. sales by Toshiba and the Norwegian firm, Kongsberg Vaapenfabrik, for at least two years. Similar bills are pending in the House, which has approved legislation banning sales of Toshiba products in military post exchanges.

Prestowitz, who was in Japan when the Senate acted, said Tokyo officials were almost totally occupied with the problem.

"They saw this as the event that could trigger a U.S. response that could close markets and take things out of control" in U.S.-Japanese trade relations, he said. "So they launched a major campaign. It's not just a company lobbying-PR campaign. This was taken up virtually all the time by all the top officials of the government."

Toshiba did very little lobbying prior to the Senate vote in the belief that the American lawmakers would accept the argument that its subsidiary, Toshiba Machine Co., acted independently and without the knowledge of the parent.

Within days of the vote, however, Toshiba Corp. hired former Nixon White House aide Leonard Garment and his law partner, former Oklahoma congressman James R. Jones, a Democrat, to join David P. Houlihan, a trade specialist in the Washington office of Mudge Rose Gutherie Alexander & Ferdon.

They insisted that Toshiba is not running a public relations campaign. Rather, they said, they are quietly trying to persuade key administration officials and influential lawmakers that punitive action will not be as effective as the steps the company and the Japanese government are taking to prevent future violations.

"We look at this as a political and economic problem," Garment said in an interview just after a luncheon meeting in June to discuss the issue with Defense Secretary Caspar Weinberger.

While Toshiba Corp. has avoided going public with its campaign, its wholly owned U.S. subsidiary, Toshiba America, is warning its 4,000 employes that their jobs are on the line and is telling customers that they could be hurt by a ban on Toshiba products.

"The employes are being urged to speak out," said Jones, the former congressman working with Garment. The regional office near Dallas sent out 6,000 letters in one week, although Houlihan insisted this was not an action pushed by the parent company.

He added that customers, concerned that they will be hurt if their access to Toshiba products is curtailed, are beginning to come forward.

Sen. Jake Garn (R-Utah), a prime Senate sponsor of sanctions against Toshiba and its Norwegian partner, reacted angrily to claims that his legislation puts American jobs at risk.

"I think those are scare tactics calculated to avoid legislation. That bothers me that they would resort to using threats like that," Garn said.

He said punitive legislation against Toshiba is needed "because what they did was so serious. Good heavens, we sent the Walkers {part of a U.S. spy ring} to jail for what they did."

Toshiba already has taken some lumps from the Pentagon, which refused to award it a $104 million contract for lap-top computers even though its bid was considered the best. The contract went to Zenith Corp. instead.

But Houlihan hopes that Toshiba's own efforts to build strong internal export control procedures that will serve as a model for other Japanese high-technology companies, along with the Japanese government's moves to improve its watchdog role over sales to the Soviet bloc, will deflect any further U.S. sanctions.

It remains unclear, however, whether the Japanese government regards export controls as a serious national security matter or is merely engaged in an exercise to resolve trade frictions at the smallest possible cost.

During a visit here in July, Hajimi Tamura, head of the powerful Ministry of International Trade and Industry, gained the acceptance of the Reagan administration for his government's plans to upgrade export control procedures.

But senior MITI officials immediately began trying to wiggle out of the agreement he reached with the late Commerce secretary Malcolm Baldrige and had to be reined back in by their superiors.

C. Itoh, the trading company that handled the transaction between Toshiba Machine Inc. and Moscow, jumped into the battle when Sen. Alan J. Dixon (D-Ill.) began pushing legislation placing sanctions on the trading company's activities in his country.

Roderick M. Hills, chairman of the Securities and Exchange Commission in the Ford administration, accompanied C. Itoh representatives to meet with Dixon.

Hills said he acted at first as a friend of C. Itoh officials he had gotten to know when he was president of Sears World Trade, the short-lived Sears, Roebuck and Co. venture in the trading company business. Hills said he later agreed to take on C. Itoh as a client.

After receiving a classified briefing from Robert Dean, the National Security Council specialist on technology transfer, Dixon toned down his bill, making it apply to all trading companies involved in illegal sales to the Soviets instead of aiming it specifically at C. Itoh.

"He felt the best way to proceed was with a generic bill," said William Mattea, Dixon's legislative director.

"The bottom line is that the senator decided to make minor changes in the legislation," added Charles Smith, Dixon's legislative specialist on defense issues.

Dixon also toned down his speech introducing the bill, deleting portions that called C. Itoh "the most aggressive" of all Japanese trading companies in soliciting Soviet business and naming major U.S. defense contractors that C. Itoh represents in sales to Japan's Defense Agency.

"Why should the United States trust C. Itoh to deal with our major military secrets in light of its actions in the Toshiba diversion and its ongoing aggressive courtship of Soviet business?" Dixon said in a July 27 draft of his speech, made available to The Washington Post. The sentence was deleted when Dixon delivered the speech on the Senate floor on Aug. 7.

This was likely due to the Dean briefing, which reportedly showed no conclusive evidence that C. Itoh officials knew they were involved in an illegal deal transferring sensitive technology to the Soviets.

Hills insisted that all C. Itoh did was handle the financing and shipping, which involved a falsified bill of lading, and had nothing to do with servicing the milling machines.

Hills also arranged for Deputy Secretary of State John C. Whitehead to assure reporters inquiring about C. Itoh's role that "the United States doesn't have any information that implicates C. Itoh" in knowingly breaking export control laws.

NEC accelerated its lobbying efforts in early August, after a leading Japanese newspaper, Mainichi Shimbun, quoting Washington sources, said the giant corporation had illegally sold strategic products to the Soviet Union.

NEC and two other companies -- Mitsubishi Heavy Industries and Sumitomo Heavy Industries -- vigorously denied the accusation after the report sent their stocks plummeting on the Tokyo exchange. The Japanese embassy here also strongly protested the report.

Within days, the law firm headed by former Democratic national chairman Charles T. Manatt registered with the Justice Department as a representative of NEC. Also in the California-based firm is Thomas B. Evans Jr., a former Republican congressman from Delaware, who is expected to do most of the Hill lobbying for NEC.

Their first success was getting the Pentagon and the White House to release an unusual joint statement saying that NEC and the other companies named in the Mainichi report were not implicated in any diversions of technology to the Soviets.

But without naming any other companies, the White House acknowledged "tentative indications" that more Japanese firms "have been involved in illegal diversions."

The Japanese campaigns will intensify when Congress returns Sept. 8 from its Labor Day recess. The House-Senate conference committee will consider the trade bill that contains the Garn sanctions against Toshiba, to which the Dixon bill is likely to be added.

Staff writer Clay Chandler contributed to this report.