"What we need to do is put the offer on the table, let's get 100 percent of these flights flying and get serious negotiations off the ground again and I make that offer publicly," the union's Joint General Secretary Tony Woodley told BBC Radio 4.
A BA spokesman dismissed Woodley's comments, saying Unite had already rejected the management offer three times.
"The union has been threatening strikes at British Airways for months. Our business is being continually damaged by uncertainty, and Unite is now seeking to create more.
"We have a business to run and customers who want to be flown to their destinations," the spokesman said.
Regardless of whether or not talks resume between BA and Unite, the Conservatives are determined to broaden the issue into a vitriolic attack on Labour's links with unions.
They launched a billboard campaign entitled "Cash Gordon," featuring a picture of the prime minister clutching a fistful of bank notes behind his back.
"Unite are funding Brown's election campaign. What will he give them in return?" the billboards read.
The Conservatives have faced uncomfortable questions on their own funding since one of their biggest donors, millionaire businessman Michael Ashcroft, revealed on March 1 he was not based in Britain for tax purposes.
(Additional reporting by Kylie MacLellan and Rosalba O'Brien)