"We call upon you to initiate steps to start a dialogue immediately on the matter," BNP's acting secretary general Mirza Fakhrul Islam Alamgir said in a letter sent to his Awami League counterpart Syed Ashraful Islam.
A three-member BNP delegation handed over the letter to Ashraf, who immediately called Alamgir to acknowledge its receipt.
"Thank you very much (for the acknowledgement)...We hope you will come forward (to end the impasse) as we have," Alamgir told Islam.
Awami League leaders said the party would brief media on its response to the BNP proposal tomorrow.
More From This Section
In an address to the nation last week, Hasina suggested the formation of an all-party interim set-up for election oversight. Zia yesterday rejected Hasina's proposal and floated a formula for forming a non-party caretaker regime.
Zia also ruled out the possibility of contesting polls, which are to be held by January 25, 2014, if Hasina remained as head of the government.
The proposals from the Awami League and BNP came amidst growing uncertainty about the fate of the election and mounting tensions between the two parties.
The Awami League, which has a three-fourths majority in the current parliament, scrapped the caretaker system by amending the Constitution two years ago. It acted after the Supreme Court ruled that the system was contrary to the Constitution.
The opposition organised violent protests against the decision, leaving scores of people dead in recent months. The BNP has repeatedly contended that polls will be fair only under a non-party government.