In my opinion, it will have one and that only for E-W or E-W to/from N-S.
The downtown I-24/40/65 loop is by no means any kind of bypass and but it is the only full freeway loop in Middle TN.
I-440 is billed as a bypass of I-40 downtown, but as it is only 4 lanes (2-2) and is frequently more congested than I-40 itself. Plus, its east and west ends are well within rush hour congestion.
Briley Parkway (TN 155) is a half-loop on the north side. (or 2/3 if you count the partially at grade section between I-40 and I-24 on the southeast side of town.) In recent years, it has been upgraded between I-40 on the east side and I-65 to make it 6 or 8 lanes instead of 4 and got rid of many substandard freeway features. You CAN use it as an I-40 bypass, but as it is so much longer, it is only faster during the worst of rush hour. If you are willing to use the at grade section, you can do I-24. I-65 - no dice.
Briley and I-440 kinda sorta make a full loop but as you need to use I-40 on both sides to go between I-440 and Briley, you still run into rush hour congestion.
So, that leaves the soon to be completed TN 840 south loop. It may work as an I-40 bypass if I-40 is congested enough that the extra distance on 840 won't hurt you. But, it will likely remain much as it is now, a connector between I-40 and interstates 24 and 65 on the south side of town. So, I guess it is a bypass if you're making those movements.
Now, I'm arguing from a practical perspective. From a technical standpoint, I'd say the answer is 3 as I'd count I-440 and Briely separately due to their lack of connections except via I-40 or surface streets.