This is one of those questions that depends entirely on who you mean or how you’re defining “Austria.” If the question is whether Emperor Franz Joseph was interested in joining the German Empire in 1871, then the answer is “not unless Germany was prepared to make him the emperor.” But by 1871, it was already quite clear that the emperor would be chosen from Prussia, not Austria. That Prussia and Austria had gone to war in the 1860s but Prussia had made no territorial claims against Austria proved Bismarck’s intention to pursue a Kleindeutschland without Austria rather than a GroßDeutschland that included it.

If the question regards whether Austrians generally wanted to join the German Empire then it really depends on whom you ask. Generally, politics among German-speaking Austrians by 1871 could be identified along three or four lines: liberals, who had formed the majority of national governments since 1848 and were against Anschluß (union with Germany) if it meant Austria taking an inferior role); socialists, who tended to view the national question as secondary to economic issues but would have accepted Anschluß if only to build a stronger working class; German ethnic nationalists, led primarily by Georg Schönerer, who absolutely wanted Anschluß into a greater German Reich; and the Christian Social Party, who preferred Austrian independence over Prussian supremacy that they viewed as likely to suppress the Catholic clergy.

This question is a bit more interesting if you pose it in 1815 or 1848, when Prussia was not yet playing a hegemonic role in German national politics. Once Bismarck is working fully in concert with the Prussian throne, the question becomes one of whether Austria will accept second fiddle. To most Austrians, that answer is no.

Good sources on this issue include Katja Hoyer’s “Blood and Iron” on German unification and the empire. Also, Carl Schorshke’s “Fin de Siecle Vienna” has an excellent chapter on Schönerer and Kurt Lueger, who led the Christian Social Party into power in the 1890s.