The answer is "no".

Consider function $$f(x,y)=\sqrt{x^2+y^2}+|y|$$ Note the $v_0=i$ is a subgradient at $(0,0)$. The gradient of $f$ is defined if $y\ne0$ and at all these points its first coordinate is strictly less than 1. Hence the statement follows.

The answer is "no".

Consider the function $$f(x,y)=\sqrt{x^2+y^2}+|y|$$ Note that $v_0=(1,0)$ is a subgradient at $(0,0)$. The gradient of $f$ is defined if $y\ne0$ and at all these points its first coordinate is strictly less than 1. Hence the statement follows.