The Gemara in masechet Shabbat (86-88a) discusses the exact date of matan torah, and the gemara's discussion of what the torah reading and the haftarah are for shavuot reflect choices which point out the commemoration of the Sinaitic revelation (Chabakuk, which Rashi connects to matan Torah, for example). Also, Shmot 19 does point to the third month (Sivan) as the zman of matan torah.