<p>We consider the compressible Navier-Stokes system in three dimensions with general inflow-outflow boundary conditions, meaning that we prescribe a boundary velocity which has a negative normal component and accordingly the density is prescribed on the inflow part of the boundary. We establish a blow-up criterion in a class of strong solutions in the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L^p-L^q\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mi>p</mi> </msup> <mo>-</mo> <msup> <mi>L</mi> <mi>q</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> framework. In particular assuming the boundedness of the quantities <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\((\varrho ^{-1}, \varvec{u})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msup> <mi>ϱ</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> <mrow> <mi mathvariant="bold-italic">u</mi> </mrow> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and of a suitable norm of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\nabla _x \varrho \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">∇</mi> <mi>x</mi> </msub> <mi>ϱ</mi> </mrow> </math></EquationSource> </InlineEquation> the solution remains regular and the blow-up does not occur. We develop the condition on <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\nabla _x \varrho \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">∇</mi> <mi>x</mi> </msub> <mi>ϱ</mi> </mrow> </math></EquationSource> </InlineEquation> because we need a new approach in order to accommodate the inhomogeneous boundary conditions, as the standard estimates on the material time derivative work when the normal component of the boundary velocity is zero.</p>