Budget director James C. Miller III told Congress yesterday that the Reagan administration might accept a relaxation of the statutory deficit goal for fiscal 1988 and urged lawmakers to pass an extension of the national debt ceiling unencumbered by amendments strengthening the Gramm-Rudman-Hollings balanced-budget law.

Miller's remarks to the Senate Budget Committee represented a significant retreat for the administration and underscored the political problem facing the White House as the Democratic-controlled Congress moves toward a legislative scenario designed to force Reagan to accept a tax increase, which he has vowed to veto.

The president, in a series of speeches, has repeatedly insisted that Congress should adhere to the Gramm-Rudman-Hollings deficit target of $108 billion for fiscal 1988, which begins Oct. 1.

Congressional Democrats have joined the drive to restore the Gramm-Rudman-Hollings automatic spending-cut mechanism as a means of forcing Reagan to accept a $19.3 billion tax increase. A provision in the original law that forced automatic spending cuts if Congress did not reach its budget goal legislatively was struck down by the Supreme Court last year.

Democrats hope that Reagan will abandon his veto threat if the only alternative is a harsh system of across-the-board budget cuts that would slash deeply into defense spending.

With the aid of Republicans who also favor the automatic device, the Democrats plan to attach the measure to must-pass legislation raising the national debt limit that is expected to reach the Senate floor today. The amendment would also scrap the $108 billion deficit target for fiscal 1988 that many lawmakers feel is unattainable, and replace the balanced-budget law's fixed yearly deficit targets with a requirement that the deficit be reduced $36 billion each year.

The Democratic strategy has put the Reagan administration, which has a history of supporting a new Gramm-Rudman-Hollings "trigger," in a politically delicate position. Miller hinted today that he might recommend that Reagan veto the trigger -- and the increase in the debt limit along with it -- if the bill does not also include other budget revisions giving the president increased authority over spending decisions and requiring Congress to adhere to its budget blueprints.

That could set off a high-stakes confrontation between the White House and Congress because at midnight Friday the national borrowing limit will fall from $2.3 trillion to $2.1 trillion, a reduction that could cause an unprecedented U.S. default on its security obligations by the end of the month.

In addition to insisting on a $108 billion fiscal 1988 deficit, the president two months ago negotiated an agreement with conservative Republican senators who agreed to go along with a 60-day extension of the debt limit in response to a pledge that the White House would support their efforts to toughen the balanced-budget legislation when the debt ceiling lapsed this month. Miller yesterday said it would be "dangerous" not to pass a clean debt limit.

Miller said the administration would "view with favor" an easing of the deficit target as long it is part of a package of budget revisions, many of which are unacceptable to Congress.

He also said that restoration of the Gramm-Rudman-Hollings trigger should be accompanied by the other budget changes.

"A Gramm-Rudman fix without budget reform is unwise," he said. "It robs the president of the tools he needs to negotiate an enforceable budget."