President Reagan announced yesterday that he will propose legislation to establish educational savings accounts and a voucher system for giving federal aid for education directly to parents of poor children so the parents can use the money at a public or private school.

The president said his latest proposals, coupled with a bill asking Congress to approve tuition tax credits worth as much as $300 for parents of private school students, will heighten competition for students and excellence in education among the nation's public and private schools.

"These proposals will expand opportunities by allowing parents to keep more of their money rather than taxing it away to finance bigger educational bureaucracies," Reagan said in his weekly radio address. "They will also increase healthy competition among schools. Without a race, there can be no champion, no records broken, no excellence in education or any other walk of life."

In the Democratic response, Rep. Harold Washington (Ill.), upset winner of his party's Chicago mayoral primary last month, said Reagan's claim of concern about education is "misleading" because his administration has cut federal funding for education.

Washington also urged Congress to support a nuclear-freeze resolution scheduled to be voted on this week. Reagan has opposed the freeze, saying it will reward the Soviet Union for its recent nuclear arms buildup and leave the United States at a disadvantage.

Washington said the federal government should be "rechanneling federal expenditures from guns and bombs to books and enlightened minds."

Reagan reiterated his view that the nation's education system is ailing and said the solution is to abolish the Department of Education.

"We must do a better job of teaching the basics," Reagan said, "insisting on discipline and results, encouraging competition and, above all, remembering that education does not begin with Washington officials or even state and local officials. It begins in the home where it is the right and responsibility of every American."

Reagan has been making a show of concern for the nation's schools recently. He visited in Florida last week with a group of prize-winning science and mathematics students and said the United States is in danger of falling behind West Germany, Japan and the Soviet Union in scientific advancement unless the nation's educational quality improves.

The president has tied his concern about schools to a call to permit prayer in public schools.

Yesterday, he asked listeners to support his proposed constitutional amendment to permit voluntary prayer in public schools. He said that in addition to the problem of increasing government intervention in education and falling scholastic-aptitude test scores in the last 20 years, today "even God, source of all knowledge, has been expelled from the classrooms."