Boober said he has been told Dunlap took a large amount of pills after his girlfriend, Jennifer Leigh Dodson, died.
Dunlap is under 24-hour surveillance by deputies at the hospital, Boober said.
"He's alert and talking on the phone and those types of things," Boober said.
Other than giving his condition, City Hospital spokeswoman Teresa McCabe could not elaborate on Dunlap's treatment Tuesday.
Dodson, 20, of 308 New Oak Tree Court in the Willow Spring Farm apartment complex, was found dead inside her apartment around 9 a.m. last Thursday.
The apartment complex is off Prospect Avenue, outside Charles Town's city limits.
Dodson's throat had been cut. Her sister and two friends found her body lying face down on her living room floor, Boober has said.
The women had gone to the apartment to pick up a 2-year-old child Dodson was baby-sitting. Dodson's 1-year-old daughter also was found inside the apartment, unharmed in her crib, Boober said.
Boober said friends and family members told police that Dodson and Dunlap were having a sexual relationship, but that Dodson had been trying to end it.
Dunlap, 45, of 312 Williams St., was hospitalized last Thursday morning after he was found slumped over the wheel of his pickup truck, which was locked and parked at the Princess Street boat ramp in Shepherdstown, W.Va., police said.
Inside the truck, police found two large kitchen knives, one of which had blood on it, police said.