Current carrying wire is placed near permanent magnet so wire experience a force, the Lorentz force but what is the nature of this magnetic interaction between wire field and magnet field? Is this a repulsive force? attractive force or both at the same time? (wire is being pushed from one side and pulled from another)
You no doubt are happy with the idea that the north pole of one magnet repels the north pole of another magnet.
The two forces on the magnets are a Newton third law pair - the force on magnet $1$ due to magnet $2$ is equal in magnitude and opposite in direction to the force on magnet $2$ due to magnet $1$.
The forces of interaction are due to the magnetic field between the magnets.
In your case the force on the current carrying wire wire due to the magnet is equal in magnitude and opposite in direction to the force on the magnet due to the current carrying wire with the magnetic field providing the interaction between the wire and the magnet.
The force on the magnet is usually not noticed as the magnet is "fixed" (eg via frictional forces) to the ground.