Eight Republican governors, joined by the Reagan administration's secretary of transportation, are to ask President Reagan today to move immediately to negotiate a voluntary trade agreement with the Japanese that would limit automobile imports significantly. a

Well-placed White House sources said Reagan has made no decision on the issue, which has provoked one of the sharpest internal battles of his young administration.

One source said the weight of the internal argument appeared to tilt toward those who favor an import limitation to help the hard-hit domestic auto industry, but that the strong opposition of Office and Management Budget director David A. Stockman has left the issue in doubt.

Known to favor such an agreement are Transportation Secretary Drew Lewis, who heads a task force that is to make a formal recommendation on the issue next week, Commerce Secretary Malcolm Baldrige and Labor Secretary Raymond J. Donovan.

The "free trade" opposition, led by Stockman, includes Treasury Secretary Donald T. Regan and Murray Weidenbaum, chairman of the Council of Economic Advisers.

Somewhere in the middle is Bill Brock, the president's special trade representative, and Secretary of State Alexander M. Haig Jr., who in words of one White House official, "has yet to show his hand."

Reagan has held one meeting on the issue where both sides strongly expressed their views. A decision is considered likely next week, after another White House meeting presided over by Reagan.

The governors are to press for the import agreement in their White House meeting today. They also are to seek a new 10 percent tax credit, one year of depreciation of hand tools instead of the two years now allowed, and a 20 pecent tax credit for research and development.

The group is chaired by Gov. William G. Milliken of Michigan, the state where the impact of Japanese automobile imports has been felt most strongly.

The other governors are James A. Rhodes of Ohio, James R. Thompson of Illinois, Christopher S. Bond of Missouri, Lee S. Dreyfus of Wisconsin, Richard L. Thornburgh of Pennsylvania, Robert D. Orr of Indiana and Pierre S. du Pont IV of Delaware.

The governors are expected to point out to Reagan that European countries have limited the import of Japanese cars and argue that the U.S. industry deserves similiar protection.

Historically, Reagan has been a free-trader. But during the election campaign, while in Michigan, he promised to help the auto industry, and aides indicated he favored a voluntary agreement.

There is some feeling in the White House that the Japanese are more receptive to such an agreement than they have ever been.

While no specific numbers have been agreed upon, the betting in the White House is that the target would be between 1.4 billion and 1.6 million Japanese cars a year. Presently, the import level is 1.9 million cars a year.

In this week's Cabinet-level meeting Reaan argued that there was no need for an import agreement because the U.S. auto industry was expected to make between $3 billion and $4 billion in profits this year.

This figure was derided by advocates of the agreement, and Treasury officials subsequently acknowledged that their numbers were wrong and that the industry stands to have profits of no more than $1 billion, if that.

Haig, as he often does, has preserved his options. One source said Haig's neutrality was a good sign for advocates of the import agreement because he had been expected to oppose it.

But another source said Haig has been impressed by Stockman's opposition to any import limitations, which would have the effect of increasing automobile prices for U.S. consumers even while they helped the industry.

At a breakfast meeting with reporters Wednesday, Brock described the Reagan trade policy as "a free-trade policy tinged with patriotism."

And Baldrige said that the American auto industry needed "some running room," adding that it will take from three to five years for it to get into a competitive position with the Japanese.

There is growing protectionist sentiment in Congress for some sort of limitation on auto imports, but it is considered unlikely that legislation on mandatory quotas would pass if the president opts to seek a voluntary agreement.

That decision is now expected no later than the end of next week.