The appeal came as Awami League of Prime Minister Sheikh Hasina and Bangladesh Nationalist Party of Leader of Opposition Begum Khalida Zia visibly remained stubborn on their stance over the electoral system.
BNP is demanding restoration of a now scraped non-party caretaker government system for election oversight, while the League suggests the poll time administration will comprise elected representatives of major parties with the incumbent premier at the helm.
"What the way forward looks like is for the parties of Bangladesh to decide. However, non-violence is essential to any solution," it said.
Secretary of State, John Kerry, has written letters to Hasina and Zia, encouraging them to engage in constructive dialogue on a way forward, the statement said.
In his letters, delivered to them by the US Ambassador Dan Mozena, Kerry had expressed concern over political situation in the South Asian country, which is now headed to polls.