From the Fellowship of the Ring:
Neither he nor the Balrog was killed by the fall, and Gandalf pursued the creature for eight days until they climbed to the peak of Zirakzigil. Here they fought for two days and nights. In the end, the Balrog was cast down and it broke the mountain-side as it fell.
And from the Two Towers:
at last he fled into dark tunnels. They were not made by Durin's folk...Far, far below the deepest delvings of the Dwarves, the world is gnawed by nameless things.
This sounds to me like the Balrog was fleeing from Gandalf. Why did Gandalf chase him for such a lengthy time to the top of Zirakzigil and risk his life?
Didn't he think that maybe it was more important to ensure the safety of the Fellowship and to make plans with Galadriel in Lorien? The Balrog was already awakened by the Dwarves many years ago but never cared to leave Moria. I think that the Fellowship's encounter with Durin's Bane did not alter Sauron's knowledge of the Balrog either, so I don't think Gandalf chased it down only because he didn't want Sauron to make use of the Balrog..