Boil, steam, or microwave the whole mangoes until soft. Cool, then remove the peel and inner 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, 270 degrees F (135 degrees C). Stir in the mango pulp, add the saffron threads, if desired, and boil until the mixture thickens, about 5 minutes.
Pour cooked jam into sterilized jars and seal according to canning directions.