In Looking toward Ararat : Armenia in modern history by Ronald Grigor we see Lavrentiy Beria asking Stallin to claim some parts of eastern Anatolia.

This was around 1950~ if memory is correct. 5 year prior to this, Soviet-Turkish Treaty of Friendship and Neutrality ended and wasn't renewed by SSCB's decision, Turkey was keen to renew the treaty.

Soviet's Montreux Convention opposition is well known too. Right when Soviets informed Turkey they wouldn't renew the neutrality treaty, Turkey and Soviets went through the so called Turkish Strait Crisis, in which the Soviets repeatedly pressured Turkey for the renegotiation of Montreux Convention.

Strait Crisis actually lasted for a few decades. Between 1930~ to till Stalin's death (1953). Marshall Plan was accepted in Turkey in 1947-8.

So, to summarize, Marshall Plan was accepted by Turkey at a time when Soviet pressure was being felt all over. Straits, claims on Anatolia (Kars - Ardahan). By the time Stalin died and was replaced by Nikita Khrushchev, which didn't follow Stalin's stance on Turkey, Turks had already choosen their side.

Please keep in mind that this wasn't just about geopolitics too. It was actually about economics itself too. Turkey's economy was one of the worst in Europe. In fact, despite receiving economic help from Britain and despite the Marshall Plan, country went through the 1946 economic crisis, 1954 economic crysis and eventually had to go for IMF and devalue its currency. Such an economic help couldn't come from the Soviets.

Would they be invaded? Hard to answer. I think not though. Because even if Turkey didn't rely on West, if a possible war were to break out between Soviets and Turkey, Western support (esp. from Unites States) would come pouring in very quickly.

To answer your question with a TL;DR | I don't think Soviets could be an alternative to the West. I don't think Soviets would attempt to invade Turkey.