The possibility that Sumerians originated from north Mesopotamia raise the probability that Proto-Sumerians had the occasion to contact the Fertile Crescent farmers who moved to South Caucasus and further north of Caucasus.

 The possibility that the Sumerians originated from northern Mesopotamia increases the likelihood that Proto-Sumerians had contact with Fertile Crescent farmers who moved into the South Caucasus and beyond. This could mean that some lexical parallels between Sumerian and Proto-Indo-European (PIE) are genuine. Aleksi Sahala from the Helsinki Institute summarized proposed parallels. Not all cases are of high quality, but some have strong chances of being real cognates.

Here are some examples:

Sum. gud, gu4 ‘ox, bull; cattle’ ~ PIE *gwou(s)- ‘cow; ox’; Hitt. *kuṷāu-;
Sanskrit: go; Greek: bous (βοῦς); Tocharian B: keŭ; Old Norse: *kú; Armenian: kov.
Cattle were domesticated in the core Fertile Crescent region of West Asia and spread to the South and North Caucasus. It is plausible that the term used by pastoralists in the Pontic and Caspian steppe came from northern Mesopotamia, as it did for the Sumerians. The Egyptian words ka ‘ox’ and kaut ‘cow’ may share this origin. Notably, ancient Egyptian farmers carried T1a1a and H2 haplotypes, which were also present among historic Armenian and South Caucasus farmers.

Sum. šáḫ(a) ‘pig; boar’ → Akkadian šaḫû ‘pig’; Ugaritic šeḫû ‘pig’ ~ PIE *suh₁- ‘swine’; Sanskrit: sūkara (सुक्र); Tocharian B: suwo; Latin: sūs; Gothic: swein.
Variants include šaḫ, šúḫ (ŠUBUR). The reading with is widely accepted and supported by Akkadian. Similar words are found in Kartvelian languages, e.g., GZ ešw- ‘wild boar, pig’. These likely share a prehistoric etymology. The Armenian xoz is not directly derived from PIE but may be related to the same areal term or borrowed from Parthian.

Sum. sí-sí ‘horse’ ↔? Akkadian sisium ‘horse’ ← Hurrian issi(a) ‘horse’ ~ PIE *h₁ekwos ‘horse’; Hittite: aśuwas; C. Luwian: a-aš-šu; Sanskrit: áśva (अश); PIE Anatolian: *aĉwa-; Latin: equus; Albanian: sasë; Armenian: eš.
The trajectory of this word is likely from PIE to Sumerian. The author conjectures Indo-Iranian mediation, but the Armenian eš / išoy, formerly meaning ‘horse’, is the best source for the Hurrian form (Petrosyan 2002). The Armenian term later underwent a semantic shift (Martirosyan 2009). Sumerians and Akkadians likely received the word from Hurrians. Most Sumerian attestations are from the Ur III period, after the Akkadian period.

Sum. urud(a) ‘copper’ → Akkadian erû ‘copper’ ~ PIE *h₁reudh-ó- ‘red’; Sanskrit: rudhira; Avestan: raoðita; Tocharian A: rtär; Greek: eruthros (ἐρυθρός); Lithuanian: raudonas; Gaulish: roudos; Old Norse: rjóðr.
This is another notable example. The semantic shift is probably from a color term to a metal, similar to argentum ‘silver’. This word is attested in Sumerian from the 4th millennium BCE. Copper use predates Mesopotamia in the Highlands. If the word for ‘red’ was originally an Indo-European term, the most likely period is the Chaff-Faced Ware culture (Late Chalcolithic, 4200–3500 BCE), a mixed population almost certainly including IE speakers.

For other linguistic parallels, see the link in the comments. Also relevant are studies on PIE–Semitic lexical parallels.