"Send names of your party lawmakers, they will be given portfolios whichever you want...Let us join hands to stage a peaceful and credible election," Hasina said in a meeting with Awami League's nomination seekers.
She urged Zia to join the polls process as there were doubts if the Bangladesh Nationalist Party (BNP) would join the polls under a government led by Hasina.
"Only BNP did not join the all-party cabinet. Therefore, I request the opposition leader to send lawmakers to join the cabinet," Hasina said.
A BNP-led 18-party opposition coalition with fundamentalist Jamaat-e-Islami being its major partner has been spearheading a street campaign demanding formation of a "non-party" neutral government for overseeing the polls.
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The coalition has called the "all-party cabinet" a farce, saying elections under Hasina would not be "credible".
However, Alamgir denied the reports even as a number of sources on both sides of the political divide confirmed their meeting over the polls-time government system as they spoke to the media separately.
"I am sorry. I cannot make you (journalists) happy. There was no meeting between us," Alamgir told reporters laughing off the report as they approached him for comments.
"I have also come to know that the meeting was held but there was no notable progress," A S M Hannan Shah, BNP's highest policy-making standing committee member, was quoted as saying.
Even as Awami League remained tight-lipped about the talks and Alamgir said the "rumour" was spread to "create confusion", Information Minister and president of ruling grand alliance's partner Jatiyo Samajtantrik Dal Hasanul Haque Inu said "the meeting took place indeed".