The brain’s limited information processing capacity leads to competition among concurrent inputs at the neuronal level. Selective attention is a cognitive process that resolves this competition by filtering out irrelevant sensory information while enhancing the neural representation of relevant stimuli. This phenomenon is well-explained by the theory of biased competition, which posits that attention biases neural processing towards behaviorally relevant stimuli when multiple objects are present. However, the mechanism of attention remains under debate. We show how biased competition can be solved in a biophysically plausible spiking neural network by biasing the recurrent network dynamics towards the attended stimuli through top-down attention signals. We trained a recurrent spiking neural network following Dale’s law to solve a classification task while receiving two simultaneous input streams and an attention cue. Our biologically realistic networks successfully replicate several experimental observations. We show that neurons develop mixed selectivity for stimulus identity and input stream and that attentional effects on individual neurons strongly depend on their selectivity. Consistent with prior research on macaque neurons in area V4, we demonstrate that neuronal responses to stimulus pairs more closely resemble the weighted average of individual stimulus responses than their sum. Networks in which top-down attention signals exclusively modulate recurrent activity perform at least as well as models where attention signals have a driving role. Overall, our model suggests that classification and attention can be learned simultaneously, allowing the network to flexibly switch between inputs.