"close with" means: A British military and historical phrase meaning to engage in direct combat with an enemy — to move in close enough to fight hand-to-hand or.... It carries a violent or aggressive reference, so it is worth noting the context. In most cases the word is used jokingly or for emphasis rather than as a sign of real intent. A calm question about where it was heard is usually all that is needed; pay more attention to behaviour and tone than to the word itself.