Investigators who searched the downtown hotel room of John W. Hinckley Jr. confiscated two tape recordings, one of which includes a conversation between Hinckley and a woman believed to be Jodie Foster, the teen-aged movie star Hinckley was apparently trying to impress when he allegedly shot President Reagan, according to informed sources.

The other tape recording taken from Room 312 of the Park Central Hotel here, the sources said, was of Hinckley playing the guitar.

Law enforcement officials are considering the recorded conversation as further evidence of an obsession with the actress, an obsession that investigators believe led the shy, 25-year-old drifter to shoot the president.

An unmailed, two-page letter also found in the hotel room -- addressed "Dear Jodie," signed "John Hinckley" and apparently written less than two hours before Monday's shooting -- refers to telephone conversations with her.

"Although we talked on the phone a couple of times, I never had the nerve to simply approach you and introduce myself," the letter reads.

"Jodie," the letter reads, "I would abandon this idea of getting Reagan in a second if I could only win your heart and live out the rest of my life with you, whether it be in total obscurity or whatever . . . The reason I'm going ahead with this attempt now is that I just cannot wait any longer to impress you."

Foster, an 18-year-old freshman at Yale University, said in a news conference Wednesday that she had never talked with Hinckley, although numerous letters signed "John W. Hinckley" or "J. W. H." were among the thousands of pieces of unsolicited mail she has received.

Investigators are working on the theory that the conversations referred to in the letter may have been made by an anonymous caller that Foster never knew by name. That caller could have been Hinckley, and the taped conversation could be one of the calls, the sources said.

The tapes are now in the custody of the FBI, which is evaluating them as part of its investigation into the shooting that wounded Reagan, presidential press secretary James S. Brady, D.C. policeman Thomas K. Delahanty and Secret Service agent Timothy J. McCarthy. All are recovering in downtown hospitals.

Federal investigators are checking the long-distance telephone calls from every place Hinckley is known to have stayed, including the home of his parents in a fashionable suburb of Denver, Colo.

The FBI has declined comment on all aspects of anything recovered from Hinckley's hotel room here. The FBI inventory of those items is said to number more than a dozen pages, but the usually public record of that inventory has been sealed by the court.

Meanwhile, a task force of some 50 people, principally FBI agents complemented by Secret Service officers and D.C. policemen, is sifting through hundreds of leads in an attempt to trace Hinckley's travels across the country in the months prior to the shooting.

Hinckley has been charged with attempted assassination of the president, a charge that could result in life imprisonment upon conviction, and assault on a federal officer -- the Secret Service Officer.

A federal judge yesterday tentatively declared him mentaly competent to stand trial, but ordered more thorough mental tests, including some to determine if he was sane at the time of the shooting. Hinckley is now being held in a psychiatric wing of the Federal Correctional Institute in Butner, N.C.

John Hinckley Sr., chairman of the board of Vanderbilt Energy Corp. of Denver, and his wife visited their son for about two hours yesterday, along with lawyers from the Washington firm of Williams & Connolly, sources said.

Before he was flown to Butner by helicopter Thursday, Hinckley was being kept isolated in the Quantico Marine Base brig under extraordinarily tight security precautions.

Hinckley appeared passive to persons who saw him there. His sole request during his confinement was for an electric razor. Law enforcement officials promptly bought one for him, according to informed sources.

Investigators believe that Hinckley developed an obsession for Foster, whose most talked-about role was that of a teen-aged prostitute in the 1975 film "Taxi Driver."

In that film, the driver, a mentally unstable war veteran who has fallen in love with the prostitute, stalks a political candidate and plans to assassinate him. But those plans are foiled by a security guard.

Hinckley was arrested at the Nashville metropolitan airport on Oct. 9 when three guns he was carrying in his suitcase set off a metal detector. Then-president Carter was in town on the same day and then-candidate Reagan was scheduled to be in Memphis on the same day.

The plane Hinckley was trying to board when he was arrested was headed for New York -- where both Reagan and Carter were scheduled to be the following week.

Federal investigators now believe there are definite parallels, sources said, between the behavior of Hinckley and that of Arthur Bremer, who shot George Wallace in Laurel, Md., in 1972 after stalking the candidate around the country. The author of the screenplay for "Taxi Driver" said the story was inspired by Bremer's life. It is still unclear, however, whether Hinckley ever saw the film.

What is clear, from the letter and the findings of investigators, is that Hinckley pursued Foster for seven months. He followed her to Yale, bragged to bar patrons that he was her boyfriend, hand-delivered numerous notes to her doorstep and stood outside her dormitory until, the letter says, he became "a topic of more than a little conversation, however full of ridicule it may be."

Foster said at her press conference that the letters she received at Yale that presumably came from Hinckley "were assumed to have been love-type letters."

In one instance, she said, she received a greeting card with the words, "I love you," written several times. Foster said she threw away the first letters she received, but several received after March 1 were turned over to her college dean and are now in the possession of the FBI.

The Hinckley family has said little about their son since his arrest. On Thursday, their next-door neighbor, William Sells Jr., told reporters that the Hinckleys were unaware of their son's reported infatuation with Foster or his possession of guns until it was reported in the media.

Investigators have found evidence that Hinckley purchased at least six guns, three of which were confiscated in Nashville and two others of which were the same type of cheap, .22 caliber revolver used in the shooting of Reagan.

Federal investigators now have five of the guns in their possession, including one found at the scene of Monday's shooting outside the Washington Hilton Hotel, sources said yesterday.

On the day of the shooting, Hinckley's parents said that their son had been under psychiatric care for five months before the shooting.

Law enforcement officials have located the psychiatrist, who is practicing in the Denver area, but he has refused to discuss Hinckley with them, citing the confidentiality of his relationship with his patient.

Meanwhile, Hinckley's defense lawyers told government prosecutors yesterday that they would not appeal the federal court order committing their client for psychiatric examination, a procedure that is expected to last at least 90 days.