I think you have a confusion about the magnetic force on a current carrying wire and the magnetic field of this wire. Based on my assumption, I will try to disentangle this confusion. If you have a current carrying wire and you put this wire in an external magnetic field, you can deduce the magnetic force that the source of this external magnetic field will act on your wire. Simply, apply the the right hand rule for the magnetic force direction on a positive charge such that your thumb point in the direction of the charge velocity and your four fingers point in the direction of the applied external magnetic field then the direction of the magnetic force is perpendicular to your right hand palm. If the charge is negative, then the magnetic force will be in the opposite direction (pointing out from the back of your right hand).
For the direction of the magnetic field that your current carrying wire will produce, point your right hand thumb in the direction of the current and curl your fingers around the wire. The direction of the curl is the direction of the magnetic field. For example, in your mentioned situation, if the wire is carrying a current into the page, then the magnetic field will be in concentric circles around the wire in a clockwise direction.