Kennedy's death occurred at what some people may consider the height of the Cold War, with the Cuban missile crisis, the Bay of Pigs invasion and such events.
Oswald was a self-proclaimed "supporter" of Communism who went to live in the USSR and even brought home a Russian wife. Despite the convenience of placing the blame on a Soviet plot behind the assassination, this potential option was never explored, neither formally nor informally by US investigation and other authorities.
Why?
Naturally, because of the existing Cold War conflict, in the immediate hours following the assassination the Americans did actually genuinely consider the possibility that the Soviets were behind the assassination, and also feared that the assassination may have been part of a wider attack on the US or government officials:
The possibility that "the shooting was part of a far-ranging conspiracy that had not yet run its course was in the thoughts of everyone."^1
Indeed, Charles Roberts (Newsweek's longtime White House correspondent) recalled "a lot of people thought, as I did, of Lincoln's assassination, where not only Lincoln, but four or five of his Cabinet were marked for assassination, that it might be, just might be, an attempt to literally wipe out the entire top echelon of government."^2
Lyndon Johnson's initial thoughts were also of Soviet involvement: "What raced through my mind ... was that, if they had shot our president ... who would they shoot next? And what was going on in Washington? And when would the missiles be comin'? And I thought it was a conspiracy, and I raised that question. And nearly everybody that was with me raised it."^3
To answer your question, though, it was partly because the Soviets were considerably agitated to persuade the Americans that the Soviet government had nothing to do with JFK's assassination and disavowed Oswald's actions. I actually wrote an answer on a similar question which I'll copy and paste:
The immediate reaction from the Soviet government was sympathy and cooperation. The Wilson Center Digital Archive has some great primary documents concerning the Soviets' immediate response here, including the letters approved and sent by Soviet Premier Nikita Khrushchev to President Lyndon Johnson and JFK's widow Jackie Kennedy, offering his sympathies and condolences. Khrushchev's successor, then Chairman of the Presidium of the Supreme Soviet of the USSR, Leonid Brezhnev also sent a message to LBJ offering condolences, and Khrushchev's wife Nina also sent a personal message to Jackie Kennedy.
Khrushchev's letter stated that he viewed the assassination of the late President as "heinous" and and that it had "evoke[d] the indignation of the Soviet people towards the perpetrators of this vile crime."
These documents also reveal that the Central Committee held a meeting on the day after the assassination to discuss their response to the Kennedy assassination, the summary of which was titled "On Measures [to be Taken] in Connection with the Murder of U.S. President J. F. Kennedy" and how to deal with the funeral. They appear to have concluded that their steps should be to send telegrams offering condolences, such as the ones sent to LBJ and Jackie Kennedy, as well as telegrams instructing Anastas Mikoyan and other Soviet diplomats to Washington to attend JFK's funeral.
Another document, which you can view here, provides a telegram sent from the Soviet Ambassador to the United States Anatoly Dobrynin to Moscow requesting advice concerning the news that Lee Harvey Oswald, who had communist connections, had been arrested and had been reported to have spent time in the USSR. Dobrynin states that after checking, it appeared the Oswald had indeed spent time in Minsk and had a Russian wife. Dobrynin notes that the media were focusing "more and more" on Oswalds left-wing sentiments and connections. Dobrynin suggested that the Soviets may wish to cooperate with the US government in providing Oswald's correspondence with his Russian wife.
In a telegram to Moscow from Andrey Gromyko, Gromyko stated that "slanderous fabrications" were being made by the US media with regard to Oswald's connections to the Soviet Union and Cuba. Gromyko suggested that if the US government requested information on Oswald's stay in the USSR, the embassy in Washington could provide them with a report. The reply from the Central Committee in Moscow approved this suggestion.
In all correspondence between Moscow and Washington, the Soviets were sympathetic and condolent. Soviet diplomats in Washington were worried that Oswald's connections with Cuba and the USSR would cause the US to believe the Soviets were involved in JFK's assassination, and the correspondence between these diplomats and Moscow suggest that the Soviets were keen to distance themselves from the assassination, and wanted to cooperate so as to ease tensions inevitably caused by Lee Harvey Oswald's communist connections.
References