Mr Kennedy, the eldest son of Senator Robert Kennedy, who was assassinated in 1968, believes that vaccines may cause autism—a view that is shared by Mr Trump.
The two met earlier this week, although Mr Trump's spokeswoman Hope Hicks said that no final decision had yet been made about his appointment.
At a Republican primary debate in 2015, Mr Trump said he was in favour of vaccinations, but was concerned that there were safety issues—including autism—that needed to be resolved. He said that several of his own employees had complained their child had had a fever and became autistic after a routine vaccination.
During his campaign, Mr Trump met Andrew Wakefield, the gastroenterologist who suggested a link between the MMR (measles-mumps-rubella) vaccine and autism and was later struck off by the UK's General Medical Council. Afterwards Mr Wakefield said: "For the first time in a long time, I feel very positive about this, because Donald Trump is not beholden to the pharmaceutical industry."