President Reagan has decided to set up a new mechanism inside the White House to thrash out policy on such legal issues as tax-exempt status for private schools and the Equal Rights Amendment, on both of which the administration has been politically embarrassed in recent weeks.

The president's senior advisers had been debating for some time establishing a Cabinet council on legal affairs, similar to the five other such councils that already exist. But they did not act earlier because of opposition from Attorney General William French Smith, administration sources said.

Smith was reluctant to bring legal issues over to the White House and submit them to discussion, these sources said, but failures of coordination on the tax exemption question and on the ERA have overcome the attorney general's objections. Smith will be chairman of the new Cabinet council just as the principal Cabinet officers involved chair the other five established last Feb. 26.

The new council, which is not yet in operation, will enable administration officials to bring political and other considerations to bear on legal issues before decisions are made.

"There is high political--and for that matter policy--content in a lot of these issues," one administration official said.

Until now the Reagan White House has tried to consider the policy and political aspects of legal decisions at the daily senior staff meeting, but that roughly 25-minute session is primarily arranged to discuss events of the day and make sure that presidential advisers know what their colleagues are working on.

"You couldn't properly treat with legal matters in those meetings. There wasn't time," one official said.

The legal affairs council will discuss such questions as whether the Justice Department should file amicus briefs in court cases; what positions to take on appeals; and what sort of testimony the government's officials should present to Congress on legal questions. Legal issues arising in Cabinet departments other than Justice also will be brought before the Cabinet council.

For months, some critics of the Reagan administration have accused the Justice Department of operating without coordination with the White House.

When Justice and the Treasury Department announced Jan. 8 that the administration was reversing 12 years of policy in order to grant tax exemptions to private schools that discriminate, they touched off a storm of criticism and brought a two-step damage control effort.

First, Reagan announced that contrary to what his representatives had said Jan. 8, he would submit legislation to deny tax exemptions to segregated schools. The bill was quickly drafted and sent to Congress together with notice that Reagan had instructed the IRS not to process applications for tax exemptions pending congressional action--except he approved a loophole permitting such exemptions for two schools.

Only a few days earlier, the government had produced confusion by sending apparently contradictory messages on the Equal Rights Amendment. First, Justice surprised observers by announcing it would appeal a judge's order striking down the ERA ratification process as unconstitutional. Reagan opposes the ERA.

The next day, Justice added that, although it would appeal, it did not think speedy Supreme Court action appropriate.

Officials acknowledged that both these recent cases contributed to the decision to establish a legal affairs council.

The other Cabinet councils are on commerce and trade, natural resources and environment, economic affairs, human resources and food and agriculture.