Since transcendental numbers $\alpha \in \mathbb{C}$ are the same as complex numbers which are not algebraic over $\mathbb{Q}$, and since algebraic numbers (algebraic number fields, etc.) are very properly the domain of modern-day number theory, it stands to reason that aspects of transcendental number theory are tightly interwoven with algebraic number theory.

(Questions which are seemingly just about ordinary rational integers are quite typically clarified by considering them as elements of larger algebraic number fields. For example, the question of when a prime is a sum of two squares is immensely clarified by working in the context of the Gaussian integers $\mathbb{Z}[i]$.)

Transcendental number theory interfaces quite a bit with analysis as well. For example, the classical proofs that $e$ and $\pi$ are transcendental involve a little number theory and a lot of evaluation of special integrals. I wouldn't have classified Schanuel's conjecture as a problem of analysis per se (without a sense that I was somewhat distorting matters), but you could definitely place this (and transcendental number theory in general) at an interface between algebraic number theory and analysis (and quite a few other fields as well, for example model theory and logic).