President Reagan yesterday put off a decision on whether to impose restraints on shoe imports after his Cabinet split in recommending tariffs or initiating a case charging foreign companies with unfair trade practices, administration sources said.

Cabinet members meeting with the president yesterday afternoon were unable to give him a unanimous recommendation because of divisions between those who want to get tough on imports and others who oppose trade restraints, sources said.

The departments of Agriculture, Commerce and Labor and the U.S. Trade Representative took what was considered the hard-line position, favoring increasing the 8.8 percent tariff on nonrubber footwear, sources said.

Those supporting higher tariffs reportedly argued that if the administration didn't stand tough on the shoe question, at a crucial period when the protectionist fervor on Capitol Hill is running high, Congress would pass harmful legislation to cut off imports.

The Treasury and Transportation departments, the Office of Management and Budget and the National Security Council favored filing a Section 301 case against foreign shoemakers, a position characterized by some sources as virtually "doing nothing."

Under Section 301 of the Trade Act of 1974, the U.S. Trade Representative would initiate a case charging foreign shoemakers with unfair trading practices. The USTR would investigate the case and make a recommendation to the president, who would have a wide range of options to remedy the problem.

The shoe case originated with a Section 201 case filed by the Senate Finance Committee. Section 201 of the 1974 trade law involves cases filed with the International Trade Commission, an independent agency that investigates such petitions and makes recommendations to the president.

White House spokesman Larry Speakes said that the president probably would make a decision in several days.

The shoe case has gained significance because some legislators feel it will give them an indication of how tough the administration will be in protecting domestic industries. Congress has complained that the administration is not doing enough on the trade issue as the U.S. merchandise trade deficit, a record $123.3 billion last year, has run at an even faster pace this year, totaling $70 billion already. The administration estimates the 1985 deficit will reach $150 billion.

Forty senators recently sent a letter to Reagan requesting that he impose quotas, and saying that the credibility of the U.S. trade laws "is clearly at stake in your treatment of this case.

"For those of us who do not represent major footwear constituencies, the importance of your granting a sufficient level of import relief in this case goes much further than the plight of this industry alone," the senators wrote. "We all represent constituents who are increasingly faced with severe competition from imports in sectors as varied as agriculture, heavy machinery and high technology."

But 19 other senators urged the president to reject restrictions on imported footwear, claiming that it would cost consumers millions of dollars.

The ITC recommended in June that the president impose a novel import quota system in which the government would auction the right to import certain amounts of shoes. The administration does not favor quotas, sources said, because foreign companies simply would raise prices to make up for lost volume and reap higher profits.

Under tariffs, the U.S. government would get the benefit from duties imposed on imports. Foreign shoe companies still might raise prices, but probably by smaller amounts to avoid pricing the shoes out of the market.

The president is required under law to decide the issue by Sept. 1. The ITC recommended an 18 percent reduction in imports of nonrubber footwear valued at more than $2.50 a pair. Imports of these shoes would be reduced from 575 million pairs in 1984 to 474 million pairs in the first year of the quotas.

The ITC also said that the quotas should remain at the same level the second year, but gradually increase in the final three years of relief from import competition.

The domestic shoe industry and some shoe-state congressmen said they favored quotas over tariffs because tariffs do not prevent imports from being sold here.

Employment in the domestic shoe industry has dropped from 215,000 workers in 1970 to about 120,000 today. Opponents of import restraints contend that they would cost American consumers between $50,000 and $80,000 for each job saved in the industry, in which the average annual wage is about $14,000.

Another issue debated by the Cabinet has been the effect of import restrictions on the ability of some developing countries to earn enough foreign exchange from exports to pay the interest on their debts.