On the basis of first-principles calculations, we investigate transport properties of spin-polarized electrons in defective metallic single-wall carbon nanotubes (SWCNTs) under homogeneous transverse electric fields. Either vacancies or carbon adatoms are introduced in (10,10) SWCNT and are shown to play a role of quasi-localized magnetic impurities. The applied transverse electric fields change the relative position of the energy levels of the defects with respect to the Fermi energy so that the spin-polarized conductances are shown to be tunable. For some impurities, the orientation of the majority spin electrons in conducting channels at the Fermi energy can be switched to the opposite spin by an experimentally attainable electric field. Our results suggest that pure carbon or organic nanomagnets could be realized in SWCNTs, and their spin transport properties are controllable by transverse electric fields.