The search for 12 jurors to hear the case of John W. Hinckley Jr., charged with trying to assassinate President Reagan, may be completed late today, finally setting the stage for the presentation of evidence.

Court officials said late yesterday that 36 prospective jurors have been found qualified for a seat on the jury and 16 have been excused after private questioning over the last three days. Actual selection of 12 jurors and six alternates will begin as soon as a pool of 43 to 45 eligible persons is assembled, court officials said.

That process may be complete by the end of the day, and presentation of evidence could begin as early as Monday.

Meanwhile, U.S. District Judge Barrington Parker announced he will hold an unusual Saturday hearing to address critical but still unresolved questions about evidence to be used.

One major issue to be discussed is the defense claim that testimony from government psychiatrists is "tainted" and should be excluded because those doctors learned of evidence concerning Hinckley that the court later ruled had been obtained illegally.

Hinckley contends that he was legally insane at the time of the shooting because a mental illness prevented him from either abiding by the law or understanding the wrongfulness of his acts. The testimony of psychiatrists from both sides will be critical to the case.

Parker also has yet to decide whether the government should have to prove beyond a reasonable doubt that Hinckley was sane, or whether the defense should have to show by a preponderance of evidence that Hinckley was legally insane.

The issue of so-called tainted testimony arises from the psychiatrists' interviews of FBI agents who conducted a 30-minute interview with Hinckley about four hours after the shooting. Parker ruled that the FBI interview of Hinckley violated his rights because no lawyer was present and his statements could not be used.

The defense says that the agents' observations about Hinckley's demeanor, later relayed to the doctors, were "absolutely critical" to the psychiatrists' eventual conclusion that Hinckley had the mental capacity to abide by the law. At the least, the defense wants any testimony based on the agents' observations excluded.

Even though other statements made by Hinckley immediately after his arrest have not been excluded, the prosecution is equally eager to get Hinckley's demeanor at the FBI interview into the case because they believe it will be an additional counterpoint to the insanity defense.

Assistant U.S. Attorney Roger M. Adelman told Parker that the information from the interviews was a "speckle on the whole picture," which the doctors derived from "a vast amount of other evidence," including interviews with Hinckley himself.

At Saturday's hearing, three of the government's doctors will be questioned in court to determine if their knowledge of the suppressed evidence affected their opinion.

Four government doctors are expected to testify, including forensic psychiatrist Jonas Rappeport of Baltimore; Parke Elliot Dietz, the head of Harvard Medical School's Institute for Law and Psychiatry, and psychiatrist James Cavanaugh, of Chicago's Isaac Ray Center at the Rush Presbyterian Hospital.