The diplomatic standoff over a request by the State Department to interview 16-year-old Andrei V. Berezhkov, the Soviet diplomat's son who may be trying to defect to this country, intensified yesterday. Soviet officials refused the request and further accused U.S. authorities of trying to prevent another Soviet teen-ager from boarding an overseas flight at Dulles International Airport.

Soviet officials charged that FBI agents, staking out flights at Dulles Friday evening, prevented a Soviet journalist, his wife and 15-year-old son from boarding a flight to Paris and "attempted to persuade" the youth "not to return to the Soviet Union."

The embassy statement released yesterday afternoon said the family was permitted to board only after the youth stated "that his only wish was to return to his homeland as soon as possible." A Soviet spokesman said a formal protest had been filed with the State Department and charged that the incident shows that a "provocation campaign is being waged against" Soviet representatives in Washington.

A State Department spokesman responded, "We strongly deny any effort on the part of the U.S. government and its officials to entice or harass any Soviet diplomat or citizen here in the U.S." The spokesman refused to elaborate, saying, "We are not going to comment on any specific incidents."

Meanwhile, the whereabouts of the Berezhkov youth remained a mystery, as it has been since Thursday when letters, signed with his name, were received by the White House and The New York Times. The letter to the newspaper said, "I hate my country and it's sic rules and I love your country." (The White House did not reveal the contents of the letter it received.)

The diplomatic difficulties were exacerbated by the fact that the youth is the son of a prominent diplomat, Valentin M. Berezhkov, who is a first secretary at the Soviet Embassy and the sole representative here of an influential Soviet think tank on American affairs.

Yesterday, Oleg M. Sokolov, the No.2 official at the Soviet Embassy here, met for about 30 minutes at the State Department with Richard Burt, the assistant secretary of state for European affairs. Neither the State Department nor Soviet officials would discuss what happened at the meeting.

But as Sokolov left the building, he answered "yes" when asked by reporters whether the Berezhkov youth is still in this country. He replied "no" when asked if Soviet officials would permit the interview that the State Department has been insisting on since learning of the two letters.

Meanwhile, the events set in motion earlier continued to escalate into a potentially nasty diplomatic incident.

An order by the Immigration and Naturalization Service preventing the boy's departure from the U.S. remains in effect, according to an INS spokesman, who said the order had been issued at the State Department's request. Secret Service agents and other federal authorities continued to stand guard at the Soviet Embassy's compound on Tunlaw Road in Northwest Washington, where most embassy personnel live. Authorities in unmarked cars also staked out the area around the Berezhkov family's apartment in Montgomery County, at 4450 S. Park Ave. in Friendship Heights.

Victor F. Isakov, another high-ranking Soviet Embassy official, asserted yesterday that his country considers the State Department's actions "a gross violation of international law" and said that "such an attitude" could "bring just negative consequences for all the sides."

In an interview on Cable News Network's "Newsmaker Saturday," Isakov said the incident "is the business of the family. . . . It's no business for a foreign government to say to the parents what to do or not to do."

A White House spokesman said President Reagan has asked to be kept informed on developments but is "leaving the handling to the State Department."

The incident began Wednesday when the Soviet Embassy reported to the State Department that the youth had taken his family's car and was missing. But Thursday morning, the State Department was informed that the boy had returned home. U.S. officials later learned of the two letters, a spokesman said.