The magnetic field line passing through the centre of a circular current carrying conductor is
straight line
In a circular coil, the field lines near the coil are circular because the field is mostly affected by only a small part of the conductor that is close to the concerned point. As we move towards the centre, the field is due to the entire conductor, as every point on the conductor is equidistant from the centre and the sum of the effects of all these points is such that the field becomes nearly a straight line.