to negotiate (with someone about); to talk over (something with); to tackle someone (about); to haggle; to bargain
to throw on each other; to splash (water) on each other; to pour on each other
担当者に直接掛け合った。
I negotiated directly with the person in charge.
値引きを求めて業者と掛け合っている最中だ。
I'm in the middle of negotiating a discount with the supplier.
困ったことがあれば担当部署に掛け合うしかない。
If there's a problem, there's nothing for it but to negotiate with the relevant department.