Plasmonic resonances in metallic nanostructures can strongly enhance the emission from quantum emitters, as commonly used in surface-enhanced spectroscopy techniques. The extinction and scattering spectrum of these quantum emitter-metallic nanoantenna hybrid systems are often characterized by a sharp Fano resonance, which is usually expected to be symmetric when a plasmonic mode is resonant with an exciton of the quantum emitter. Here, motivated by recent experimental work showing an asymmetric Fano lineshape under resonant conditions, we study the Fano resonance found in a system composed of a single quantum emitter interacting resonantly with a single spherical silver nanoantenna or with a dimer nanoantenna composed of two gold spherical nanoparticles. To analyze in detail the origin of the resulting Fano asymmetry we develop numerical simulations, an analytical expression that relates the asymmetry of the Fano lineshape to the field enhancement and to the enhanced losses of the quantum emitter (Purcell effect), and a set of simple models. In this manner we identify the contributions to the asymmetry of different physical phenomena, such as retardation and the direct excitation and emission from the quantum emitter.