They would also forego a salary rise until February 2011 to save the company millions of pounds.
Unite put forward the offer as part of a series of pay and productivity proposals which officials said would achieve "substantial" savings.
Sources said that union officials were surprised that the company had not accepted the offer immediately.
A deadline for agreeing thousands of job cuts and a two-year pay freeze as well as changes to conditions passed without agreement on June 30 and further talks have been held to try to break the deadlock.
Unite will hand out letters to BA shareholders at the airline's annual meeting in London next week seeking their support for an agreement.
The letter says that unions realise the difficult times airlines such as BA are facing because of the global recession, escalating fuel prices and worsening trading conditions in premium markets.
The airline made a loss of £400 million last year but the shareholders will be reminded that BA posted record profits of £883 million the previous year.
The letter warns shareholders that proposed changes from the company cut deep into operations and could eliminate any hope BA has of returning to its position as the "world's leading airline".
Unite will tell shareholders it has tabled major proposals which would reduce costs by hundreds of millions of pounds a year and provide greater flexibility and more efficient working including the pay cut as well as thousands of job losses.
"We cannot understand why the company would dismiss such significant savings and productivity measures as a mutually-acceptable, non-conflict way forward.
"Our fear is it can only be because BA's management are opportunistically using the recession to force through changes which are more far-reaching and damaging to BA's future."
Unite said warnings by BA's chief executive, Willie Walsh, that the company was fighting for its survival were "misguided" because they "talked down" the airline.
Shareholders will be told that the company wanted to introduce poorer contracted employment, cut 4,000 jobs, freeze pay for two years, introduce limitless, indefinable "total flexibility" and end lifetime career opportunities at the airline in favour of low-paid, short-term employees working for no more than five years.
"This is not our vision for BA. This is a nightmare, one which will see BA reduced to scrapping at the lower end of the labour market.
"Instead of being an employer proud of how it treats its employees, it will become one with only the bare legal minimum of protections in place.
"The reputation damage of these changes will be profound and irreversible."Reuse content