The $31 billion package to stimulate the economy that President Carter sent Congress with such a sense of urgency in January has become bogged down in political bickering amid some signs the economy is reviving by itself.

Not a single piece of the package has cleared Congress, though the House passes all of it before the end of March. The tax bill is caught is a Senate fight over how much, if any, tax stimulus is still needed. The local public works bill to create 160,000 jobs has been held up for seven weeks by a House-Senate fight over water pollution. An appropriation bill to pay for it all has been delayed in the Senate until these issues are cleared away.

The centerpiece of the program was to have been a one-shot $10 billion tax rebate to step up purchasing power. Two weeks ago the President dropped the rebate as no longer needed economically. It was also considered in trouble in the Senate, which last week agreed to delete the rebate but went against the President's wishes in retaining $4.1 billion in business tax credits. There was talk of a possible veto.

Sen. Edmund S. Muskie (D-Maine), Senate Budget Committee chairman who strongly supported the $50 rebate and was angered when the President suddenly pulled the rug out from under it during the Easter recess, has scheduled hearings with the administration's top fiscal officials today to ask their justification for the action. Testifying at 2 p.m. will be Treasury Secretary W. Michael Blumenthal, Director Bert Lance of the Office of Management and Budget, and Chairman Charles L. Schultze of the President's Council of Economic Advisers.

On Wednesday the Senate will vote on a Republican proposal for a permanent individual tax cut costing about $10 billion a year as an alternative to the $50 rebate. A similar proposal was narrowly defeated in the House, which approved the rebate.

The $4.1 billion in business tax credits that remain in the bill, along with the individual tax relief, would give industry the option of taking an increase in the investment tax credit to encourage plant expansion or a credit for hiring workers above normal needs to reduce unemployment. The President opposed both now that the tax rebate has been dropped. The House voted 4 to 1 for the jobs credit over the investment credit and in a House-Senate conference the investment credit would almost certainly be dropped.

The administration testified last week that it had dropped the tax rebate because industrial production, housing starts, gross national product and other economic indicators were up more than expected while there were some inflationary signs such as, the upward movement of wholesale prices. Unemployment was down from 7.8 per cent in December to 7.3 per cent and expected to drop to 7 per cent by the end of the year.

But the administration was still pushing hard for enactment of the jobs part of the stimulus package which is designed to put about 1 million people to work.

A House leadership staff member said White House liaison people called half a dozen times last week asking for help in freeing the $4 billion local public works bill. Finally, House Speaker Thomas P. O'Neill Jr. (D-Mass.) and Senate Majority Leader Robert C. Byrd (D-W.Va.) leaned on their conferees and they started moving toward agreement last Friday. They may agree on a bill today.

The House passed the public works bill in February. It was to be a quick starting program to put people to work by making grants for local projects that are ready for work to begin though critics say public works programs are slow-starting at best, and no use as a quick-fix for a sluggish economy.

The Senate later added $9 billion to the bill in grants to build local sewage treatment plants, part of a continuing program to clean up the nation's streams. This set off a fight with the House, which does not want to put more money into the program until the Senate agrees to make policy changes to speed the cleanup. Environmental groups fear House changes will do ecological damage.

The likely solution will be stopgap water treatment funding and some extensions of deadlines and moratoriums or regulations to give time to work out comprehensive water pollution legislation. That would free the public works money for the building season now beginning in the north.

Last year Congress provided $2 billion to start the local public works program. This was enough to fund only 2,000 of more than 25,000 applications for the grants. That money is only now coming out of the processing pipeline and becoming available for construction projects. The $4 billion in the pending legislation is to be given out for other approved projects on that original list of applications.

The President also asked for an increase of 415,000 in public service jobs, providing services not now available in state and local governments, and funds to provide 346,000 jobs in a variety of youth, veterans and Indian programs.