He described the move as being an example of the Scottish Government's "investment in our young people and the future of NHS Scotland".
The apprentices will initially be recruited to the administration, IT and facilities management sectors of the NHS. They will work to provide support to frontline clinical staff.
Mr Neil announced the modern apprenticeships during a visit to Dundee's Ninewells Hospital, where he said: "Our commitment to 500 new modern apprentices represents an investment in our young people and the future of NHS Scotland.
"I am pleased to be here today to see for myself some of the excellent work being done by modern apprentices in NHS Tayside
"Recruiting modern apprentices not only provides real opportunities for our young people, it also helps to ensure that NHS Scotland has a workforce with the skills and expertise to continue delivering first-class services, both now and in the future."
NHS Tayside chairman Sandy Watson said the health board has a "long history of training modern apprentices and was the first board to recruit modern apprentices in care in 2012".
He added: "All of our time-served apprentices have gone on to secure full-time positions with NHS Tayside, whether that be in care, administration or in one of the many trades we support.
"We are delighted that today's announcement supports the further placement of potential new recruits to the healthcare system."