The Tur (OH 27) writes that the straps should be in front of a person. As noted by the Beit Yosef there, this is stated by the Rosh (Halakhot Ketanot: Hilkhot Tefillin: 11).
It is also stated by Abudirham (Birkhot HaShahar) and Ri al-Nakawa in Menorat HaMaor (Kadmon) (ch. 2. pg. 71).
The potential Talmudic source for this is Menahot (35b) which states:
רבה קטר להו ופשיט ושדי להו. רב אחא בר יעקב קטר להו ומתלית להו. מר בריה דרבנא עביד כדידן
Rabbah used to tie the knot and allow [the straps] to fall straight down and throw them. R. Aha b. Jacob used to tie the knot and then plait [the straps] together. Mar the son of Rabina used to do according to our custom.
Rashi (s.v. Rabba) explains that this is referring to what they would do with the straps of their head tefillin that extended past the knot. Although it is a little difficult to understand, his explanation of Rabbah's practice seems to be (cf. s.v. Rabbah and s.v. k'didan) that Rabba pulled the straps to the other side of his head; i.e in front of him, and let them hang down.
This explanation of Rabba's practice is stated explicitly by Tosafot thee (s.v. `Avid).
Rambam (Hilkhot Tefillin 3:12) writes that the straps should be long enough to reach roughly the navel, although he doesn't state that they should be there; just that they should be long enough, it could certainly be inferred that they should be there, just as he writes that strap of the arm tefillin should be long enough to wrap around the arm and hand, which he holds one is supposed to do. See additionally Hagahot Maimoniyot and Kessef Mishneh there.