In his weekly radio address, Mr. Clinton again told GOP lawmakers that he's reaching out to them - to seek an agreement on Medicare and even tax relief.
Mr. Clinton described the package he vetoed as "just too big, too bloated." He said the GOP package would have put "too big a burden on our economy and run the risk of higher interest rates and lower growth."
"At the same time, they're still not providing nearly enough for education and other vital priorities," he said. Among programs he said would have been endangered were expansion of Head Start, a tutoring reading project, provision of Interconnect connections to the nation's classrooms and others.
Still, he insisted he had every reason to veto the Republican tax cut bill this week - repeating that the package is too big and too bloated. And he claims the GOP plan depends upon budget gimmicks.
Mr. Clinton noted that on the day he vetoed that plan, a panel in the the Republican-controlled Congress approved a measure that would undermine the administration's education plans.
Mr. Clinton says this would eliminate a call for 100,000 new teachers, and it would not expand or improve Head Start programs.
Despite the harsh criticism, Mr. Clinton insisted that Republicans should work with the administration to produce a budget compromise.
CBS News White House Correspondent Bill Plante reports that both sides knew all along that this tax cut never had a chance.
Meanwhile, House Speaker Dennis Hastert urged President Clinton to say he "made a mistake" by vetoing a $792 billion tax cut passed by Congress.
The Illinois lawmaker said in the weekly Republican radio address his colleagues did not want to make "a fool's choice" by accepting a smaller tax cut at the cost of more federal spending.
The Republican package called for a sweeping, 10-year tax cut as well as accelerated repayment of $2.2 trillion of federal debt, Hastert said.
Addressing the president directly, Hastert said: "Mr. President, I think you made a mistake by vetoing this tax fairness package."
Treasury Secretary Lawrence Summers said large tax cuts, like the plan favored by Republicans, could fuel inflation, drive interest rates higher and jeopardize the continued growth of the U.S. economy.
Hastert, in the House leadership job for about a year, said Mr. Clinton "may insist on tax increases this fall so that he can pay for more Washington spending. Maybe that's why he vetoed our tax relief measure."
"Some have said that we should negotiate with the president, that we should have a grand compromise where American families get minimal tax relief while the White House gets more Washington spending. That's a fool's choice. Getting minimal tax relief for a maximum increase in pending is a bad deal for the American people," Hastert said.
Republicans want to make the Social Security trust fund off-limits for anything but retirement benefits, by placing the funds in a so-called "lock box."
For decades, under so-called unified budget rules, successive American governments have borrowed a portion of the Social Security trust fund to pay for general budget expenditures.
"Mr. President, join with us in protecting the Social Security Trust Fund. It is just plain common sense," he said.