Here's a formulation of the canonical linear programming problem:
$$ \max_{\mathbf{x}}\left(\mathbf{c}^T\mathbf{x}\right) \quad\text{subject to}\quad \mathbf{Ax\leq b} \quad\text{and}\quad \mathbf{x\geq 0} $$
One can then consider the threshold variant
Given $t$ does there exist a $\mathbf{x}$ such that $$ t \leq \mathbf{c}^T\mathbf{x} \quad\text{and}\quad \mathbf{Ax\leq b} \quad\text{and}\quad \mathbf{x\geq 0}\,? $$ (Answering this question just gives a yes-or-no answer, not an actual $\mathbf{x}$ which satisfies these properties.)
Suppose this second question can be answered in polynomial time. Can $\mathbf{x}_\text{opt}$ then be approximated (each of its entries bounded by an interval of width $2^{-n}$) in polynomial time?
To start with, as long as I can ensure that $\mathbf{c}^T\mathbf{x}_\text{opt} \leq 2^{\text{poly}(n)}$, where $n$ is the total number of bits required to describe the input, then I could narrow down $\mathbf{c}^T\mathbf{x}_\text{opt}$ to an interval of width $2^{-n}$ using binary search together with a polynomial number of calls to the threshold version.
However, I don't see how to actually box in $\mathbf{x}_\text{opt}$ itself using these methods.