A statement didn't give any further details about the 29-year-old's condition.
It comes after Nelson recently missed the final of the girl group's BBC One talent show, and their hosting of the MTV European Music Awards.
Little Mix's latest album entered the UK chart at number two on Friday.
All six of the studio albums they have released since forming on The X Factor in 2011 have gone into the top five, and they have had four UK number one singles.
In a statement, the group's publicist said: "Jesy is having extended time off from Little Mix for private medical reasons.
"We will not be issuing any further comment currently and ask media to please respect her privacy at this time."
Nelson's bandmates Leigh-Anne Pinnock, Perrie Edwards and Jade Thirlwall were seen without her on the final of Little Mix: The Search on 7 November, and on the MTV EMAs the following night.
Last year, Nelson received widespread praise for discussing her mental health battle on the BBC Three documentary Jesy Nelson: Odd One Out. The documentary, which won a National Television Award, addressed body image and the impact of online bullying.