There is abundant documentation of extensive Indian Ocean maritime trade from later eras -- check out the Periplus of the Erythraean Sea, a Greek text from the mid first century CE -- but the quantity of written information drops sharply further in the past, so we are forced to look for archeological and biological evidence.

There is considerable archeological evidence for maritime trade between the Indus Valley civilization and ancient Mesopotamia -- Sargon of Akkad documented it ca. 2270-2215 BCE. But this probably isn't quite what you're looking for, seeing as how you specify Tamil in particular. Present day Tamil speakers mostly reside in southern India and Sri Lanka, rather than the Indus Valley region. (As a side note however we still don't know what language the Indus Valley civilization spoke -- one faction argues it was in the Dravidian language family... which includes Tamil!)

Black pepper (native to southern India) was stuffed in Pharaoh Ramses II's nostrils during his mummification circa 1212 BCE. Ramses II spent considerable time in Canaan as part of his campaign against the Hittite Empire. So we can speculate that there were trade links-- possibly indirect -- between South India and Egypt / the Levant. And trade can bring linguistic exchange too.

A fascinating paper published by PNAS earlier this year by Scott at al studied dental plaque from 2nd millennium BCE cemeteries in Megiddo and Tel Erani (current day southern Israel near Kiryat Gat). This plaque demonstrated

clear evidence for the consumption of expected staple foods, such as cereals (Triticeae), sesame (Sesamum), and dates (Phoenix). We additionally report evidence for the consumption of soybean (Glycine), probable banana (Musa), and turmeric (Curcuma).

Soybean, banana, and turmeric were domesticated in Southeast Asia and the Indonesian archipelago.

But they aren't the most impressive evidence of long distance Indian Ocean trade in the 2nd millennium BCE. For this we must look at the example of cloves found in an excavation of a Syrian site at Terqa dating to 1700 BCE. Cloves are from the Moluccas, in present day Indonesia.

Terqa cloves bear witness to the extraordinary range of Mesopotamia's contacts in thesecond millennium, even if they reached Syria via Harappan, Dilmunite, and/ or Babylonian middle men." (Potts, Daniel T., 1996, Mesopotamian civilization: the material  foundations,Cambridge University Press p.270)

As a final note, the Cochin Jews of Southern India claim to be descended from King Solomon's sailors and merchants.