Some Syngenta AG shareholders are angry about the Swiss agribusiness giant’s rejection of takeover proposals from rival Monsanto Co., raising the possibility that they could seek changes to Syngenta’s board to make it more amenable to a deal.

“The management of Syngenta is throwing away $15 billion to $20 billion of shareholders’ money,” said Mark Yockey, portfolio manager with Milwaukee-based Artisan Partners LP, which ranks among the top 10 Syngenta shareholders, according to Thomson Reuters data. Mr. Yockey’s estimate...