The two sides have traded barbs for weeks about whether Mr. Peltz deserves a seat on the consumer-products giant's board. Mr. Peltz's Trian Fund Management, which is one of the company's biggest shareholders with a 1.5% stake valued at $3.5 billion, has been amassing support. Another top shareholder, Yacktman Asset Management, which owns about 15 million shares of P&G or a 0.6% stake, said earlier this month it is backing Mr. Peltz. Five former board members of H.J. Heinz Co. wrote a letter to P&G's board in support of Mr. Peltz, who won a seat on the ketchup-maker's board in 2006 after a proxy fight. ISS can influence how institutional shareholders such as index-fund managers and mutual funds vote, though its recommendation isn't seen as insurmountable to the losing side. Both ISS and Glass Lewis sided with Mr. Peltz in his failed attempt in 2015 to win a seat on the DuPont Co. board.