Major freight railroads are facing pressure to add sick days for their workers from a new front: An influential investment group says some of its members are now pushing the measure that Congress declined to as part of the contracts they imposed last week to avert a potentially devastating nationwide rail strike.

Members of one union rejected a tentative deal with the largest U.S. freight railroads Wednesday, while two ratified agreements and three others remained at the bargaining table just days ahead of a strike deadline, threatening to intensify snarls in the nation's supply chain that have contributed to rising prices.