Yes...and no.

First, some background. Sino-Soviet relations had deteriorated from 1959 onward (some historians trace it even earlier to Khrushchev's Secret Speech in 1956) in the so-called Sino-Soviet Split. This split can be seen from either an ideological perspective, in which the two powers drifted apart due to a fundamental disagreement over the tenets of Marxism-Leninism; or from a geopolitical perspective, in which both sides increasingly saw each other as their main threat to state security (even over the traditional enemy of the United States). The Soviet invasion of Czechoslovakia in 1968 and the proclamation of the Brezhnev Doctrine sent alarm bells ringing in China. The Soviet Union was now claiming they had the right to intervene in any communist country not following its brand of Marxism-Leninism. To China’s leaders, this was a clear threat directed against them. A further complication was the Cultural Revolution in China. With its critique of Soviet ‘revisionism’, the mass movement had flamed jingoistic anti-Soviet sentiments among the Chinese leadership and populace.

In consideration of the above factors, Mao and other Chinese leaders decided that the Soviet Union needed to be "taught a lesson." They ordered the People’s Liberation Army (PLA) to initiate major border clashes with the USSR on the disputed Zhenbao Island on 2 and 15 March 1969. Further border clashes occurred in April and May, and on 13 August, the Soviets struck back by launching an ambush against a Chinese border patrol in Xinjiang. Yang Kui-song has shown convincingly that the Chinese intent of the clashes was to serve as a warning to the Soviet Union, and not to provoke a general war. Thus, Chinese leaders were shocked when news reached them of Soviet military, and more ominously, nuclear build-up.

It seems that there was genuine discussion among Soviet military and political leadership of a nuclear response. As Soviet archives on events in 1969 remain closed, historians can only rely on testimonies from Soviet officials. The Soviet diplomat Arkady Shevchenko claimed:

The events at Damansky (Zhenbao Island) had the effect of an electric shock on Moscow. The Politburo was terrified that the Chinese might make a large-scale intrusion into Soviet territory which China claimed...From others I heard that the Soviet leadership had come close to using nuclear arms on China. A [Foreign] Ministry colleague who had been present at the Politburo discussion told me that Marshal Andrei Grechko, the Defense Minister, advocated a plan to "once and for all get rid of the Chinese threat." He called for unrestricted use of multi-megaton bombs...Fortunately, not many military men shared Grechko's mad, bellicose stance...I talked with one of Grechko's colleagues, [General] Nikolai Ogarkov...[who] took a more realist view of the prospect of war with China...[He proposed] the alternative...to use a limited number of nuclear weapons in a kind of "surgical operation" to intimidate the Chinese and destroy their nuclear facilities...Disagreements about bombing China stalemated the Politburo...for several months.

His status as a defector in 1978 casts some doubt on Shevchenko’s account. However, the fact that a nuclear response was at the very least considered is correlated by other Soviet sources. Senior Counselor to the USSR United Nations Delegation, Valentin Karymov, stated that every kind of contingency plan was considered, including preventive strikes. Lev Deluisin, a China specialist in the Foreign Ministry, had this to say:

Discussions occurred about whether to carry out a preventive strike against all of China's nuclear complexes so as to resolve the problem...fortunately, the government rejected these options, but these opinions were expressed.

Public publications from the Soviet Union also support the idea that nuclear strikes were considered. On 28 August, Pravda, the mouthpiece of the Soviet Communist Party, published an editorial, stating ominously that:

If war breaks out under the current situation, no continent will survive the catastrophe of modern and destructive weapons and missiles.

From the American side, we know that some feelers were made to discern American attitudes to a nuclear strike. On 18 August, while having lunch with the US State Department’s Soviet specialist William Stearman, a Soviet embassy official casually raised the topic of “what the US reaction would be to a Soviet attack on a Chinese nuclear facility.” American intelligence also detected preparations consistent with a preventive strike at Soviet air bases in Siberia and the Far East during late August.

Mao and the Chinese leaders were horrified. For all their sabre-rattling, China was in no way prepared for a general war. The factional infighting during the Cultural Revolution had weakened the PLA, and there had been no specific military build-up other than some discussions within the Chinese Politburo in 1969. In August, the war scare in China reached its fever pitch. On 27 August, the CCP Central Committee issued an urgent order for the large-scale evacuation of Chinese population and main industries from big cities, while calling upon workers and residents in big cities to begin digging air-raid shelters and stockpiling everyday materials to prepare for a nuclear strike. On 28 August, an urgent mobilisation order was issued to China’s border provinces and regions. Party committees, government agencies, military commands, and ordinary citizens in provinces adjacent to the USSR were urged to be prepared for a large-scale Soviet surprise attack, while PLA forces along the Sino-Soviet border entered an emergency status of combat readiness.

Both sides recognised that drastic action was needed to stop escalation. This was achieved in a meeting at Beijing airport between Chinese Premier Zhou En-lai and Soviet Foreign Minister Kosygin on 11 September. Zhou launched an impassioned plea against nuclear war:

You say that we want to make nuclear war. In fact you are well informed about the level of our nuclear weapons [and you know that we will not do so]. You say that you will take preemptive measures to destroy our nuclear facilities. If you do so, we will declare that this is war, and that this is aggression. We will rise in resistance. We will fight to the end...You say that we want to go to war. But now we have very many domestic problems to deal with. How can you believe that we want to go to war?

This meeting effectively ended the phase of military build-up between the USSR and China, although the rattled Chinese leadership continued to issue emergency orders until mid-October 1969. It was in this atmosphere of fear that Mao started to see the USSR as China’s “main enemy”, and the seeds of a Sino-American rapprochement were planted.

Going back to the initial question, we can see there was genuine Soviet consideration of a nuclear strike against China. However, as is always in foreign relations, this was merely one card on the table, and cooler heads eventually prevailed. The matter was eventually resolved in bilateral talks between China and USSR, with little to no American involvement in the whole affair.

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Sources:

Kuisong, Yang. "The Sino-Soviet Border Clash of 1969: From Zhenbao Island to Sino-American Rapprochement." Cold War History 1, no. 1 (2000): 21-52.

Goldstein, Lyle J. "Do Nascent WMD Arsenals Deter? The Sino-Soviet Crisis of 1969." Political Science Quarterly 118, no. 1 (2003): 53-80.

Li, Danhui, and Yafeng Xia. Mao and the Sino-Soviet Split, 1959–1973: A New History. Rowman & Littlefield, 2018.