There are several researches on where was the first transcript of Rigveda written. While some say it was written in India, others suggest Syria.
According to the Indian government’s website ‘Vedic Heritage,’ The Rigveda represents India's earliest sacred book. It is the oldest and biggest amongst all the four Vedas (religious texts originating in ancient India). All the features of Classical Sanskrit poetry can be traced to the Rigveda. In it, we find the seeds of India’s religious and philosophical development.
According to a study by Penn State University, Indian grammarian Panini wrote Sanskrit's description in about 1500 B.C.
However, according to Britannica, the language originated in what is now the Punjab region of India and Pakistan. It says that Rigveda consists of a collection of 1,028 poems grouped into 10 “circles” (mandalas) and was preserved orally before it was written down about 300 BCE.
In a book written by Edwin Bryant published in 2003, ‘Quest for the Origins of Vedic Culture,’ it has been mentioned that the earliest form of Sanskrit is that used in the Rigveda was first recorded in inscriptions found not on the plains of India but what is now northern Syria. It says that ‘between 1500 and 1350 BC, a dynasty called Mitanni ruled over the upper Euphrates-Tigris basin, land that corresponds to what are now the countries of Syria, Iraq, and Turkey. Mitanni's spoke a language called Hurrian, unrelated to Sanskrit. However, each and every Mitanni king had a Sanskrit name, and so did many of the local elites’.
Since the oldest known scriptures are said to be from India, but various recent discoveries and papers are showing that the language had European and Syrian connections. However, since these things are not yet officially affirmed by experts, the language's actual origin can't be currently determined.