Federal Trade Commission Chairman James C. Miller III has urged House and Senate subcommittees to clear bills to reauthorize his "embattled" agency, but observers say a number of issues could clog the process.

The FTC's authorization expired in October 1982. The process of approving a new one has provided businesses, professional associations, consumer groups and politicians an arena for arguing over how the agency does its job.

In hearings over the past two weeks, a variety of groups have testified that the FTC's powers should be limited, expanded or left untouched. Some say the agency is too easy on business, while others think it should be easier. For instance:

* The American Medical Association and the American Bar Association said Congress should leave out any language that would give the FTC jurisdiction over professional groups. The last attempt to reauthorize the agency failed, in part, because of lobbying by professional groups seeking to exempt themselves from FTC regulation.

* The National Association of Attorneys General asked Congress to give its members power to enforce parts of the Federal Trade Commission Act, arguing that FTC enforcement has been lax.

* Representatives of independent grocers complained about alleged predatory pricing by major supermarket chains, and argued that the FTC should act to stop such activity.

* Some discount retailers criticized the FTC for allowing manufacturers and retailers to enter into agreements that they say raise prices.

* An alliance of plumbing, heating and cooling contractors urged the FTC to take action against allegedly unfair competition from regulated utilities.

* Representatives of the advertising industry asked that Congress prohibit the FTC from enforcing its rules against "unfair" ads, saying that standard is too vague.

Before testifying at a House subcommittee hearing, Miller told reporters that the witness list indicated "another gripe session, with all the old shibboleths trotted forward."

Karen Johnston, the agency's director of congressional relations, said she is "optimistic" that an authorization can be passed this year "if an agreement can be reached on the professions issue" and if Rep. James J. Florio (D-N.J.), chairman of the House Energy and Commerce subcommittee that has jurisdiction over the FTC, "is willing to exercise some leadership in moving a bill."

Johnston said "strong leadership is especially critical" to prevent the authorization bill from dying of neglect in a year when Congress may be preoccupied with hotter issues.

AZCUENAGA CONFIRMED . . . The Senate has confirmed FTC Commissioner Mary L. Azcuenaga for a seven-year term. Azcuenaga, an independent, has been a commissioner since Nov. 27, when President Reagan gave her a recess appointment.

Agency insiders say Azcuenaga's appointment has boosted staff morale, since she was an FTC staff member for 11 years, most recently as an assistant general counsel.

Azcuenaga first worked for the agency as a summer legal assistant in 1972. She received a law degree from the University of Chicago in 1973 and joined the FTC permanently the following September.

RESALE POLICY . . . The FTC has outlined its policy on resale price maintenance, stating that a manufacturer can choose to sell only to retailers who agree to charge the manufacturer's suggested retail prices. In a letter to the House Appropriations Committee, the agency also said it was legal for retailers to follow the manufacturer's price directives "in order to avoid termination."

Because of complaints from retailers, the committee had asked the FTC and the Justice Department to clarify their policies on resale price agreements.

"Resale price maintenance is per se illegal," the FTC added. "A manufacturer, however, still retains considerable control over the resale of its products. As long as it is acting unilaterally, a manufacturer may announce in advance a policy of selling only to certain types of retailers -- e.g., those who sell at the suggested retail price."

As for enforcement, the agency said it has investigated several allegations of illegal resale-price maintenance and issued a complaint in one case where it found collusion in violation of antitrust laws.

Otherwise, the FTC said, "private remedies often are most appropriate."