I recall that evil creatures like Nazgûl can sense the presence of the Ring. At any rate, the Ring having a will of its own should be "calling" to the Balrog, who being a fallen Maia can make use of its powers to challenge Sauron. I believe this is exactly what the Ring wants. The Ring did try to seduce and corrupt Aragorn and Galadriel. So it makes perfect sense for the Ring to end up in the Balrog's hand if all it wants is an easy route to Sauron.
A theory has been proposed here that Balrogs being lieutenants of Morgoth share his lust for destruction and annihilation and as such can have no use for the Ring. Because the Ring ensnares you by promising you what you desire to accomplish and utter annihilation is not a worthy goal. Therefore the Ring does not "call" to the Balrog because there is no use to which the Balrog can put the Ring's power except for mindless destruction. But I fail to find that convincing for the reason that even if the Balrog were to use the Ring for its destructive purposes, that would still bring the Ring closer to Sauron or some other powerful being who can seize it.
So did the Balrog not detect the presence of the Ring on Frodo? Why didn't the Ring make use of such a great opportunity to go to the hands of a worthy bearer? Finally if the Balrog indeed knew about the One Ring, why was it not tempted by the prospect of seizing it for its own?
The question is - Was the Balrog aware of the presence of the ring? It should be given how the Ring draws evil to itself and corrupts even the most good-intention-ed if it sees a potential it can prey on. Yet the Balrog went straight for Gandalf and didn't seem to show any interest in the Ring.