I am wondering if it is possible for an entire function $\phi$ to admit a direction $v\in\Bbb C$ such that $\phi(tv)\to0$ for $t\to+\infty$ but $|\phi|$ is constant "for a while" in that direction, that is, there exists an interval $[t_1v,t_2v]$, where $t_1<t_2$, in which $|\phi|$ is constant.
I am trying to find such an example but nothing comes in mind. I still don't know if this is possible.
No that is not possible. Without loss of generality I'll take $v=1$ (otherwise consider $z \mapsto \phi(v z)$ instead). If $\lvert \phi \rvert=1$ on some real interval then $\overline{\phi(\overline{z})} = \phi(z)^{-1}$ on that interval and since both sides are holomorphic equality holds throughout $\mathbb{C}$. In particular for all $z\in \mathbb{R}$: $\overline{\phi(z)} = \phi(z)^{-1}$, that is, $\lvert \phi(z) \rvert=1$.