The lithograph is among 140 pieces available from a range of artists that can be viewed and selected from the gallery until Sunday.
French artist Matisse was born in 1869 and died in 1954. In his later years he became bed bound and focused on creating cut paper collages.
The Dancer features similar cut out shapes, although it is a lithograph print.
Leeds City Council’s executive member for adult social care, active lifestyles and culture, Salma Arif said the lending scheme allowed art to be “accessible and available to everyone”.
She said: “Bringing these beautiful artworks into our communities gives those people taking part the chance to experience and appreciate art in a completely new and individual way and to put culture at the heart of our homes and families.”
The Picture Library was established in 1961 and includes oil paintings, watercolours, drawings, photographs and a variety of prints.
Membership allows residents to loan one artwork for up to one year or swap four artworks.