Modded from an earlier answer

Soviet aid to Iraq was never consistent and ebbed and flowed throughout the Iran-Iraq War. This 1980s saw the nadir of Soviet efforts to court the Middle East during the Cold War as their networks of strategic partnerships with various Arab states collapsed. The Soviets had hoped that their sponsorship of various nationalist movements like the Ba'ath would lead to a cooperative relationship with local communist parties that would increase Soviet influence within the region. In the 1950s and 60s, both Arab nationalists and Arab communists in the Middle East shared some ideological commonalities. They both believed in a centrally-planned economy, were leery of the Western powers, and advocated a secular modern state. The problem was as the Arab nationalist movements evolved into platforms for regional strongmen like Nasser or Saddam, for whom the independence of communist parties became a threat. Saddam's suppression Iraqi Communist Party (ICP) was similiar to contemporaneous actions against Syrian communists undertaken by Assad and earlier Nasserite precedents in Egypt. Reports from the GDR's security services show that many in the Eastern bloc were alarmed by this repression but Moscow reasoned that an open break with Iraq would undo whatever influence it had with Saddam.

The Soviets did give some assistance to ICP forces in Kurdistan and provided a safe-haven for ICP leaders in the USSR. This assistance to the ICP was tenuous and low-key, but increased as when Saddam quickly and publicly condemned the Afghanistan invasion in 1979. However, Iraq at this time began diversifying its arms suppliers and economic assistance packages. The Iraqis began courting Brazil, the FRG, China and other powers outside the control of Moscow. Saddam therefore ensured that he would have alternatives if Moscow fully broke off relations and thus he put a further squeeze on the ICP.

The outbreak of war with Iran caught Moscow quite flat-footed as its diplomatic experts on the Middle East had believed that Tudeh (the Iranian communist party) could capitalize on the chaos of the Iranian Revolution and emerge as the head of a new Iranian state. Although this was highly unlikely and KGB efforts to cultivate Soviet power in Iran bore little fruit, the Iraqi invasion spelled their death knell. The Soviets declared an embargo on Iraqi arms and refused to honor existing contracts. Iraq's military reverses in 1980/81 led Saddam to backpedal on his anti-Afghanistan rhetoric as he realized he needed Soviet arms to maintain his armies as his attempts to buy replacements on the world market depleted Iraqi monetary reserves. The Soviets in turn decided to begin resupplying the Iraqis in the Summer of 1981. Again, the guiding principle of the Soviets mirrored the same rationalization of supporting the Ba'ath in the 1970s: they had invested too much in Iraq to just abandon it. The Iraqis also continued their policy of diversifying their defense suppliers, which further lessened the Soviet's leverage. Here was where American support for Iraq (never quite as substantial as the politicized mantra "the US armed Iraq!" suggests- it was quite a crowded field of arms suppliers and other aid) became especially alarming for Moscow. Saddam's continued military incompetence meant that the Iraqis started to demand far more advanced Soviet equipment, in exchange, the Soviets demanded to examine captured Iranian weapons which were advanced Western models. But beyond this functionalist relationship, the Soviets hoped that by providing Saddam with enough armed force, they could compel Iran to a cease-fire. Once that was achieved, Moscow believed it could rebuild its influence in the region by sponsoring reconciliation between Iraq and Syria. Such aspirations proved illusory given the general collapse of Soviet power in Eastern Europe in 1989 meant the Soviets could ill-afford to continue any sponsorship of Iraq after the invasion of Kuwait.

As for the question of Iran reaching out to the USSR, the Iranians already had large stocks of US equipment from the Shah's regime. Acquiring Soviet military aid would have meant incorporating a whole new set of equipment produced to different specifications. This would have further complicated Iran's already tenuous logistical system which was already suffering from being cut off from American resupply. The Iranians looked to the Soviets, as well as the Chinese and North Koreans, to provide equipment that could be easily integrated into Iran's military infrastructure like artillery or to supplement advanced Western equipment of which the clandestine sources of spare parts was in short supply. An example of the latter was the acquisition of the ZSU-23-4 which supplement the US-built Hawk SAMs. Iranian defense policy after the war continued this trend with armaments from Soviet/Russian, Chinese and North Korean sources supplementing an indigenous defense industry that reverse engineers the equipment acquired under Shah. This system has resulted in Iran's very heterogeneous order of battle.

Sources

Andrew, Christopher, and Vasili Mitrokhin. The World was Going our Way: the KGB and the Battle for the Third World. Basic Books, 2005.

Brands, Hal. "Saddam Hussein, the United States, and the invasion of Iran: was there a green light?." Cold War History 12, no. 2 (2012): 319-343.

Murray, Williamson, and Kevin Woods. The Iran-Iraq War: A Military and Strategic History. Cambridge University Press, 2014.

Sassoon, Joseph. "The East German Ministry for State Security and Iraq, 1968–1989." Journal of Cold War Studies 16, no. 1 (2014): 4-23.

Takeyh, Ray. "The Iran-Iraq War: A Reassessment." The Middle East Journal 64, no. 3 (2010): 365-383.

Soviet and Iraq were ally's against America once and their relations were strong but then they got weak after the revolution in Iran. Because the new established regime in Iran was against America, the soviets saw Iran as a new potential strong ally but Saddam was against this revolution like America, so American-Iraqi relations got strong, But then Iranians announced that they were against soviet ideology because they consider themselves Muslims and cant ally with atheism, so the soviets after knowing their mistake acted fast and supported Iraq so they don't walk empty handed from the area with no ally's and totally lose Iraq as a friend ...