House Democrats have run into a problem in their fight to reverse President Reagan's cost-cutting in social welfare programs: a Democratic-drafted bill to fund those programs that is so tight-fisted that Republicans say Reagan probably can sign it.

The $96.1 billion appropriations bill for Labor, Education and Health and Human Services, which is scheduled for House action today, is only the latest in a string of difficulties the Democrats have encountered in getting Congress to live up to the budget it adopted earlier this year.

First House and Senate tax-writing committees indicated that they will not approve the $73 billion in tax increases that the budget demanded. Now the Appropriations committees of both houses appear willing to settle for substantially less than the budget prescribed in so-called discretionary domestic spending.

Moreover, major changes in the big automatic benefit or entitlement programs--Reagan wants cuts, the Democrats favor increases--are considered increasingly unlikely as the 1984 elections approach.

While the Democratic-controlled House has approved a series of bills authorizing expanded social welfare spending and new jobs-related programs, including a $3.5 billion public service jobs program that was passed yesterday, few are expected to survive in the Republican Senate.

Thus it is the actual spending bills, especially the big labor-human services appropriations bill, that will probably count the most in the Democratic effort to reverse Reagan's retrenchment policies.

In light of this, both the Democratic leadership and rank-and-file Democratic liberals have been pressing behind the scenes, without success so far, for a substantial expansion of the bill.

"A lot of Democrats think it's dumb," said Rep. Les Aspin (D-Wis.), a member of the House Budget Committee. "What you're doing is taking away an issue. We should be jamming it to Reagan on education," added Aspin in reference to Reagan's embrace of education as a campaign issue for next year.

The dispute could spill over onto the House floor today or later when the House takes up a stopgap "continuing resolution" to fund agencies that have not received their regular appropriations by the start of the new fiscal year Oct. 1.

The bill, as approved by the Appropriations Committee last week at the behest of labor-human resources subcommittee chairman William H. Natcher (D-Ky.), includes $31.1 billion for discretionary programs, which is $3.5 billion more than Reagan recommended. But it is $4.7 billion less than the congressional budget prescribes, meaning that it is closer to Reagan's target than to Congress' own spending goal.

Overall, for discretionary programs (as opposed to entitlement programs that are not controlled by annual appropriations), the bill provides generally the same amount of spending as Congress provided for the programs during the current fiscal year, according to a Democratic whip advisory on the bill.

To the extent that the bill falls short of the congressional budget, it is a political vindication for Reagan, who rationalized his defeat on the budget earlier this year with the argument that the real fight would come later on appropriations bills.

It is also a victory for Office of Management and Budget Director David A. Stockman, who quietly lobbied key appropriators in both houses, holding out the prospect of a presidential signature if the bill was kept within bounds from the White House standpoint.

This had appeal not only for Natcher but also for Rep. Silvio O. Conte (R-Mass.), ranking minority member on the committee, who has gotten tired of getting caught in the squeeze between Reagan and Congress on spending. Conte, like Natcher, is also tired of the stalemates that have prevented final action on the labor-human services bill in recent years, forcing its inclusion in the stopgap continuing resolution.

Natcher fended off challenges in the committee with the argument that additional funds, especially for programs that have not yet been authorized by Congress, could be provided in a supplemental appropriations bill later in the year.

Natcher has also reportedly stood firm against arguments for additional spending, especially for education, from House Speaker Thomas P. (Tip) O'Neill Jr. (D-Mass.), Majority Leader James C. Wright (D-Tex.) and Democratic Whip Thomas S. Foley (D-Wash.), among others. Out of regard for Natcher and his stature in the House, however, they are said to be reluctant to take him on publicly.

After working his will in the House, Stockman turned to the Senate.

Following a meeting with Stockman last week, Sen. Lowell P. Weicker Jr. (R-Conn.), chairman of the Senate's Appropriations subcommittee on labor, health and human services, agreed to lop $1 billion off his version of the legislation, leaving it a negotiable $300 million over the House version.

This could mean, for the first time in recent years, an agreement between both houses of Congress, and with the president as well, on the government's major domestic spending bill. But it would leave the Democrats with a lot of symbolic votes on authorization bills, without the dollars to back them up.