Today, the month of Elul is seen as being an introduction to the period of the High Holy Days (Rosh HaShanah and Yom Kippur).
For instance, Tur Orach Chaim 581:
התקינו חז"ל שיהו תוקעין בר"ח אלול בכל שנה ושנה וכל החדש כדי להזהיר ישראל שיעשו תשובה ... ויש מי שמרבין לומר סליחות ותחנונים מר"ח אלול ואילך
Our Sages instituted that we sound the shofar on Rosh Chodesh Elul every year and throughout the month to warn Israel to repent ... And there are those that increase the recitation of penitential prayers from Rosh Chodesh Elul and onwards.
I am unaware of any Talmudic source that makes this connection. Where do we first see a link made between Elul and repenting in advance of the High Holy Days?
The earliest source I see so far is a quote from R. Isaac ibn Ghiyyat (רי"ץ גיאת) who lived in the mid-11th century. He is quoted by Rosh on Rosh HaShanah 4:14 as writing:
אמר לי רב כהן צדק מנהג בשתי ישיבות לומר תחנונים בהני עשרה יומי שבין ר"ה ליום הכפורים וכן אומר רב עמרם ורב האי מנהג לומר תחנונים בהני עשרה ימים בלחוד ושמעינן דמקצת אתרוותא קיימי מראש חודש אלול
R. Cohen Tzedek said that the custom in the two yeshivot [Sura and Pumbedita] is to recite penitentiary prayers during those ten days between Rosh HaShanah and Yom Kippur. Similarly, R. Amram and R. Hai said that the custom is to recite penitentiary prayers during those ten days only. And I have heard that in some places they have them from Rosh Chodesh Elul.
I know that Pirkei de-Rabbi Eliezer 46 discusses a custom of blowing the shofar, but it seems to be only on Rosh Chodesh Elul, and is connected to Moshe ascending Har Sinai for the final time, rather than explicitly as a call to repentance.
Conversely, one could argue that the ancient custom to recite haftarot of comfort (שבע דנחמתא) throughout Elul and only afterwards to recite two hafatarot of repentance (תרתי דתיובתא) indicates that Elul was seen more in its relation to the fast of Tish'ah BeAv that preceded it, rather than to Rosh HaShanah and Yom Kippur. This custom is quoted in Tur Orach Chaim 428, referencing Pesikta de-Rav Kahana, which is (or at least contains a core that is) an Israeli midrash dating to around the fifth or sixth centuries.