Quantum critical points separating weak ferromagnetic and paramagnetic phases trigger many novel phenomena. Dynamical spin fluctuations not only suppress the long-range order, but can also lead to unusual transport and even superconductivity. Combining quantum criticality with topological electronic properties presents a rare and unique opportunity. Here, by means of ab initio calculations and magnetic, thermal, and transport measurements, we show that the orthorhombic CoTe$_2$ is close to ferromagnetism, which appears suppressed by spin fluctuations. Calculations and transport measurements reveal nodal Dirac lines, making it a rare combination of proximity to quantum criticality and Dirac topology.