When we think about whether narcissists lack empathy or not, the answer might not be always black and white. Research attempts to find out the relationship between narcissists and empathy. So, do narcissists lack empathy, or do all narcissists lack empathy?
We generally assume that narcissistic people lack empathy, and we employ this assumption to account for some of their undesirable behaviors and traits. However, there are circumstances where narcissistic individuals will display empathy, which can be confusing.
Many narcissistic people who seem to lack empathy for other humans in their lives can express enormous compassion for their pets, and they may overtly express empathy for a sad child or an injured animal. Thus, let’s consider the possibility that narcissists are consciously and unconsciously unwilling to empathize, rather than lacking the capacity to do so. If this is so, then how do we make sense of their unwillingness?
A lack of empathy is often considered to be one of the distinctive features of narcissism. However, this is not entirely the case. The criteria for the formal psychiatric diagnosis of narcissistic personality disorder in the Diagnostic and Statistical Manual of Mental Disorders of the American Psychiatric Association includes “lacks empathy,” but this designation has a critical qualifier: namely, “is unwilling to recognize or identify with the feelings and needs of others.” 
An unwillingness to empathize with another person is not the same as being unable to empathize. The notion that a person has a capacity for empathy, yet is not empathically responsive, may be useful for understanding the personality characteristics of people we label as narcissistic.
Empathy is the capacity to think and feel oneself into the inner life of another person.  Some regard empathy as a vicarious affective response based on the awareness of another person’s emotional state.
 Many definitions of empathy include the concept of perspective-taking—emotionally or cognitively seeing things from the other person’s position.
 Thus, empathy can involve both a cognitive process (the ability to understand another person’s view in terms of what the other is thinking or feeling) and an experiential process (resonating with another person’s emotional response).
Some researchers have found that the cognitive functioning necessary for empathy, such as the ability to role-play or take another person’s perspective, occurs in a different location of the brain than the emotional aspects of empathy, such as sensitivity to what another person is feeling or experiencing.
 Whether one is narcissistic or not, our brains simulate the feelings of those around us. This ability to unconsciously mimic another’s feelings enables us to reconstruct within us what other people may be experiencing.  
Given the many complicated interactions we have with others throughout our lives, the ability to automatically understand what is going on with someone else is a crucial skill for successful social functioning.
Some studies have shown a relationship between narcissism and deficient emotional empathy, but that narcissists, nevertheless, can recognize and react to the suffering of others, even if they are motivated to disregard such distress in other people. 
The capacity to empathize does not preclude its use for bad behavior or destructive purposes.   Some people may consciously or unconsciously be motivated to withhold an empathic response to control a partner, or they may exploit their understanding of another person’s emotional state to manipulate them or to gain power.
Using their empathy manipulatively, for example, people with narcissistic pathology know how to evoke insecurity in their partners and provoke attachment anxiety.