Some of the world’s largest agricultural companies are looking to combine with one another as three years of shriveling crop prices have pressured profits, in what would be the industry’s first big shake-up in at least a decade.

Syngenta AG is discussing with DuPont Co. a potential combination with DuPont’s agriculture division, according to people familiar with the matter. DuPont is also separately discussing a potential alternative agriculture deal with Dow Chemical Co., which is exploring a sale of its seed and pesticide...