The House of Representatives will begin debate today on amendments to the controversial administration-backed bill to authorize a substantial increase in the nation's contribution to the International Monetary Fund.

Congressional sources said that a vote on the entire bill probably would not come before Thursday. President Reagan has said the $8.1 billion in increased U.S. funding for the international financial rescue agency is critical to the international and domestic economies.

A similar bill passed the Senate easily last month, but opposition in the House has been strong on both sides of the aisle. House Speaker Thomas P. (Tip) O'Neill (D-Mass.) wanted to hold the bill until he was sure there were enough votes to pass it, which would probably have put off a decision until September. But yesterday, O'Neill agreed to allow the debate to begin.

"He still doesn't believe they the administration lobbyists have the votes," one source said. "But he's willing to let them prove that they do."

Treasury Secretary Donald T. Regan, who began a concerted push last week to enlist the aid of the business community to sway undecided Republicans, was optimistic yesterday. He told reporters that "we're confident that when necessary we'll have enough votes to put it over. It's still a battle."

Conservatives have opposed an increase in IMF funding because they feel it is not necessary and that the IMF can raise the money it needs on its own--either by borrowing from private lenders or using its $45 billion in gold.

Many liberals think the president should loosen up the purse strings for domestic programs before they will consider spending more money on international projects. Many legislators dislike the legislation because they think it is a bailout of major banks that made too many risky loans to troubled countries like Mexico and Brazil.

One of the amendments would require the United States to oppose future IMF loans to South Africa, because of its policy of apartheid or racial separation. The members of the Congressional Black Caucus have told O'Neill that they will vote against the IMF funding increase if this amendment--opposed by the administration--is knocked out. An aide to the Democratic leadership yesterday said the amendment was "deadly serious."

Yesterday, the Center for International Policy, a non-profit group, published a study saying that the IMF itself reported in a staff paper on May 19 that the job restrictions caused by apartheid could damage the economy in the medium-term.

The center said "the news that the IMF itself has criticized apartheid on economic grounds . . . deflects the administration's argument that the amendment will politicize U.S. involvement in the IMF." The IMF staff study was carried out on the initiative of the executive board of the IMF, one source said.

Perhaps the most widespread complaint against the IMF funding comes from legislators who see it as a bailout of major banks that made too many risky loans to troubled foreign countries.

But the IMF legislation would not affect the loans the international agency already has committed to Mexico, Brazil, Argentina and other countries over their heads in international debts. Instead, it would give the IMF the wherewithal to deal with future international financial crises.