The Education Department, which has run into stiff congressional opposition to its proposals to cut college student aid, is grappling with a more immediate crisis: The Pell grant program for low-income students is facing an $819 million shortfall this year.

At a hearing on Capitol Hill last week, deputy undersecretary Gary L. Bauer called the shortfall "a major problem" and said it "has been getting worse." Yesterday, William Dingeldein, deputy director of the department's budget service, said that although the Pell grant shortfall was not surprising, "the magnitude of it is significant."

The crisis has provoked a new round of finger-pointing on the part of the Reagan administration and Capitol Hill.

Congress has accused education officials of stalling on seeking a supplemental appropriation for the program, while pushing for some unpopular cuts in benefits to needy students. Education officials, on the other hand, say Congress aggravated the problem by passing new rules last year to make the grants more generous, but not approving enough money to pay for them.

Congress last year raised the top Pell award from $1,900 to $2,100, to cover up to 60 percent of college costs (instead of the previous 50 percent). Those changes, Dingeldein said, put the program $351 million in the red. Another $468 million was borrowed from the fiscal 1985 appropriation to cover last year's shortfall.

The administration has proposed reducing top Pell awards to $2,000, and to cover up to 50 percent of costs. Bauer and Dingeldein said earlier that no supplemental budget request was in the works. But the House and Senate Appropriations committee chairmen have warned that they shouldn't count on the program being changed -- at least not in time to affect grants for the next school year.

Efforts to solve the program come against a backdrop of jostling over the fiscal 1986 budget. The department's budget proposed cutting the Pell grant program by $644 million -- a plan that higher education groups said would eliminate 574,000 students of the program's approximately 2.5 million recipients.

But since the White House reached a budget compromise with the Senate GOP leadership, all bets are off. Said Dingeldein, "We're just sort of waiting and seeing what happens in the budget compromise." RESEARCHING RESEARCH

. . . A departmental panel is meeting to determine ways the government could improve its role in educational research.

The panel, which met once in mid-April and is to gather again the first week of June, is composed of some of the most distinguished conservative scholars in the education field. Among the members is Chester Finn, co-director of the Center for Education and Public Policy at Vanderbilt University, who is expected to be named assistant secretary of education for research. Also on the panel are Columbia University's Diane Ravitch, Joseph Adelson of the University of Michigan, Denis Doyle of the American Enterprise Institute and Michael Kirst of Stanford University.

Secretary William J. Bennett is reviewing the roles of the National Center for Education Statistics and the National Institute of Education, which is a favorite target of conservatives.

Bennett has said several times that he supports a federal role in research and statistics-gathering. Whether he eventually decides to merge NIE with NCES remains to be seen.