Homotopic maps induce the same maps on homotopy groups given path connected spaces

Question

Given pointed maps $f,g: (X, x_0) \longrightarrow (Y, y_0)$ that are pointed homotopic then $f_* = g_* : \pi_n(X, x_0) \longrightarrow \pi_n(Y, y_0)$.

Now let $X$ and $Y$ be path connected. Then if $f,g: X \longrightarrow Y$ are homotopic (not necessarily pointed homotopic) can we say $f_* = g_* : \pi_n(X,x_0) \longrightarrow \pi_n(Y, y_0)$?