Faced with strong lobbying by colleges and universities, the Education Department yesterday said it will postpone its proposed requirement that schools be made responsible for verifying that young men applying for federal student aid have registered for the draft.

Students requesting aid still would be required to declare whether they have registered for the draft, but for the first two years of the program the schools would not be required to check the accuracy of the students' claims.

If the department meets it own timetable, the rules would be in effect at the beginning of the next school year, 1983-84.

Gary L. Jones, undersecretary of the Education Department, told a House subcommittee yesterday that many of the schools, colleges and higher education associations commenting on the rules complained that verifying a male student's registration could be cumbersome and costly.

Associations like the American Council on Education, which represents 1,500 colleges and universities, had requested that implementation of the rules be delayed until after the 1983-84 school year. The Education Department refused.

However, Jones said yesterday that for the first two academic years after the rule becomes final, institutions do not have to verify whether a student reported his registration status correctly. Thereafter, verification will be required.

The Education Department's action came while the government was awaiting a final decision by a U.S. District Court judge in Minnesota on the constitutionality of the law prohibiting eligible male students who have not registered from receiving student aid.

Judge Donald D. Alsop granted a temporary restraining order against enforcement of the law two weeks ago, saying the statute was likely to violate students' rights against self-incrimination.

A hearing on a request to make the order permanent was held earlier this week. Alsop has not announced his decision.