Investigators say they learned of Lombard's alleged activities through an informant who is also facing sex charges.
According to an arrest warrant, Lombard made the child sex offer to a member of the Washington D.C. Metropolitan Police working with the Northern Virginia Regional Internet Crimes Against Children Task Force during an online chat session.
The warrant says Lombard lives in Durham with a gay partner, and that he has two adopted children.
It says he told the investigator "he could not molest the adopted 5-year-old child when his partner was around, however the partner was leaving for a four-day business trip the next day, which would allow him the ability to molest the child just as he did the last time his partner had left town."
The warrant goes on to say that Lombard admitted to drugging the child with Benadryl during the molestation. It also goes into graphic detail about the molestation - including accusations Lombard performed oral sex on the child.
In a later chat session, Lombard suggested the person he was chatting with fly to Raleigh/Durham to have sex with the 5-year-old.
FBI agents and officers from the Durham Police Department and the State Bureau of Investigation arrested Lombard at his home this week. Two children living at the home were taken to into protective custody.
Duke University said Lombard has been an employee since 1999. He has been placed on unpaid administrative leave. Duke says it's cooperating with the investigation.
If convicted, Lombard could face up to 20 years in jail.