However, BA responded it had already offered to hand back travel concessions as part of a deal.
War of words
It also accused Woodly of "negotiating through the media" rather than talking to them directly through Acas, an organisation dedicated to resolving employment disputes.
"We have already offered to reinstate travel concessions to cabin crew once all elements of our offer have been implemented," a statement from the airline said.
"We had agreed to a request from ACAS to meet on [Sunday] afternoon and are surprised that Unite did not take advantage of this," it said.
Negotiations to try and halt the planned strike were abandoned on Saturday after dozens of protesters stormed the building where the talks were being held.
Walsh has faced accusations that he is trying to break the unions at BA and Unite accuses BA of imposing changes on cabin crew and refusing to negotiate openly and fairly.
However, the BA chief executive insists that the airline is struggling to survive.
Monday's strike comes on the back of BA posting a record annual pre-tax loss of £531m [$765m] on slumping sales.
The airline, which is cutting costs and merging with Spanish airline, Iberia, in a bid to return to profitability, has been hit hard by the global economic downturn which has decreased demand for air travel.
It also faced a tough start to the current financial year due to the recent closure of airspace across Europe caused by ash from an Icelandic volcano.
BA staff are also planning two further five-day strikes starting on May 30 and June 5.