Yes and no.
On the one hand of you have a magnetic field that circulates just like it does around a steady current and the current is anything other than that steady current then the electric field will change:
$$\vec \nabla \times \vec B = \mu_0\vec J +\mu_0\epsilon_0\frac{\partial \vec E}{\partial t}$$ is the same as $$\frac{\partial \vec E}{\partial t}=\frac{1}{\mu_0\epsilon_0}\left(\vec \nabla \times \vec B -\mu_0\vec J\right)$$ so you can think of the imbalance as being associated with a changing electric field. But that electric field tends to make positive charges go that way and make negative charges go the opposite way. Which is exactly what you need to increase the current more to be what you want.
So if there isn't a wire there then the electric field just increases if there is a wire and the current is too large the electric field still changes but in a way to damp the current. And if there is too little current the electric field increases in a way to increase the current.
So having just the right current is the static solution the one that balances perfectly and let's things not change.
Now, since other answers claimed this wasn't possible I'd like to mention a specific way to create these fields.
In a wire you need a certain electric field to maintain a steady current dispute the fact that the wire is full of things inside the wire that the moving electrons bump into. You can change the current in a simple way without changing the magnetic fields that circulate around the wire.
Simply grab the wire and pull it in the opposite direction as the current. This moves a net positive charge in the right direction to decrease the current and if you keep pulling it down at a steady speed you now need a stronger electric field to have the previous current flow down the wire but that is what happens the electric field starts to increase until the right current (mobile electrons moving the same direction as you pulling the wire plus some more) is there to match the magnetic field.
You can always read Maxwell's equation as just telling the electromagnetic field how to change. The curl of the electric field telling the magnetic how to change and the the curl of the magnetic field telling the electric field how to change if it isn't absolutely perfectly matched by the current.