Boil, steam or microwave the whole mangoes until soft. Cool then remove the peel and seed. Place the mango pulp in a large bowl. Use a fork or potato masher to mash the pulp well.
Place the sugar and water in a large saucepan over low heat; stir mixture and bring to a boil. When mixture begins boiling increase heat to medium-high. Continue boiling until fine, soft threads form (135 degrees C if using a sugar thermometer).
Stir in the mango pulp and add the saffron threads (if desired) then boil until the mixture thickens; about 5 minutes.
Pour cooked jam into sterilised jars and seal according to directions.