Place chopped figs, apricots, and water in a small saucepan and bring to a boil. Reduce heat to low and simmer until fruit is plump and very soft, about 20 minutes. If needed, add just enough water to keep mixture simmering.

Remove from heat and transfer to a small food processor. Add lemon juice and salt (and port, if using); purée carefully, as the contents are hot. Add water about a teaspoonful at a time until consistency is that of a spreadable paste. Allow to cool to room temperature. Cover and refrigerate up to 1 week.