The House, acting to avert a government financial crisis that would strike by the end of the week, yesterday began moving stopgap legislation to extend the nation's authority to borrow money through Aug. 6.

The short-term extension of the $2.3 trillion debt ceiling, expected to pass the House and Senate today, would give Senate Republican and Democratic negotiators additional time to perfect a compromise to reinvigorate the Gramm-Rudman-Hollings deficit-reduction law.

The restoration of the law's automatic spending-cut mechanism is the key element in Democratic plans to force President Reagan to compromise on the budget and taxes. The restoration is to be attached to legislation providing for a longer extension of the borrowing ceiling.

The short-term measure was approved yesterday by the House Ways and Means Committee. It extends the debt ceiling of $2.3 trillion that expired July 17. The credit limit automatically reverted to $2.1 trillion -- which is lower than the outstanding federal debt and precludes the Treasury from making new debt offerings.

Since the expiration, the Treasury Department has canceled two bill auctions. If the borrowing limit is not raised by Friday, there would be an unprecedented government default because the government would be unable to redeem $10.2 billion in two-year notes.

For the Democratic-controlled 100th Congress, however, this year's debt-limit minuet also has broad political implications as the Democratic leadership attempts to use the debt legislation to force the Reagan administration to negotiate a budget accord.

Democrats, joined by conservative Senate Republicans anxious to impose additional fiscal restraint on Congress, are pushing to reinstate the Gramm-Rudman-Hollings budget-cutting mechanism that was declared unconstitutional by the Supreme Court last year. Their goal is a new provision that would be constitutional while mandating automatic across-the-board spending cuts if Congress and the White House fail to meet annual deficit goals -- which would be eased.

Democratic congressional leaders believe that restoration of the Gramm-Rudman-Hollings "trigger" is their best hope of getting Reagan to back off his opposition to the $19.3 billion tax increase that is the cornerstone of the $1 trillion fiscal 1988 budget adopted by Congress. With the trigger in place, Reagan would have to choose between the tax increase and deep reductions in the Pentagon budget.

Yesterday, Democratic leaders turned up the rhetoric, scoring Reagan for driving the nation deeper into debt and portraying themselves as the party of fiscal prudence.

"Perhaps the president doesn't know that he has run up a $1.3 trillion debt," said House Speaker Jim Wright (D-Tex.) at his morning news conference. "Nevertheless, President Reagan is still responsible for the devil-may-care, borrow-and-spend policies of his administration which have pushed our government to the brink of defaulting on our obligations for the first time in history."

Despite Reagan's earlier enthusiasm for a Gramm-Rudman-Hollings fix, administration officials have been pressing for a "clean" debt-limit extension, insisting that any revival of the balanced-budget law must be accompanied by revisions in the congressional budget process that would bolster the power of the executive branch.

The Democrats still face some hurdles, including the negotiations that resumed late yesterday in the Senate over the precise form the Gramm-Rudman-Hollings fix will take. Last week, the Senate rejected a Democratic plan and refused to let a Republican alternative come to a vote.