  We study the tachyon supertube probes in a type IIA supergravity background
which is a stringy-like G\"odel spacetime and contains closed timelike curve.
In the case of small value of $f$, which is a parameter of the background, we
use the Minahan-Zwiebach tachyon action to obtain a single regular tube
solution and argue that the tube is a BPS D2-brane. However, we find that the
fluctuation around the tube configuration has a negative-energy mode. This
means that the tachyon supertube, despite being a BPS configuration, develops
an instability in the pathological spacetime with closed timelike curve which
violates the causality.
