Given, $2x + \frac{2}{x} = 3$. Dividing by 2, we get: $$ \begin{aligned} x + \frac{1}{x} &= \frac{3}{2} \\ \Rightarrow \left(x + \frac{1}{x}\right)^2 &= \left(\frac{3}{2}\right)^2 \\ \Rightarrow x^2 + 2 + \frac{1}{x^2} &= \frac{9}{4} \\ \Rightarrow x^2 + \frac{1}{x^2} &= \frac{9}{4} - 2 \\ &= \frac{9 - 8}{4} \\ &= \frac{1}{4} \end{aligned} $$ Thus, the value is $\frac{1}{4}$.