The media storm over the deaths was reignited after an inquest this month returned a verdict of unlawful killing from carbon monoxide poisoning, with Thomas Cook criticised for having been slow to apologise to the family.

British media also criticised it for receiving more in compensation for the deaths from the hotel on the Greek island than the children's parents.

Green, who surprisingly quit as Thomas Cook CEO in November after two years successfully turning the company around, has faced criticism in the media for her handling of the incident during her tenure.

She told Reuters on Wednesday that media reports that she refused to meet the parents to apologise and that it was she who had started the process to seek damages for Thomas Cook from the group that owns the hotel over the incident were false.

Last week, current CEO Peter Fankhauser apologised to the family in person and the firm said it had donated its compensation payout to Unicef. Thomas Cook was not immediately available for comment about Green's remarks.

Green is due to receive 7 million Thomas Cook shares in July, worth around 10 million pounds ($15 million) currently, that had been part of her remuneration package.

"I have now reached out to the parents of Bobby and Christi Shepherd. On the basis that Thomas Cook are due to give me 7 million shares, I have told the parents that I will donate one third of that to a charity of their choice," she said.

Green said that while in charge of Thomas Cook she had written to the children's father to offer a meeting but received no reply.

She said the litigation process against the owners of the hotel where the children died was already under way when she joined and that she had not been aware of any compensation received whilst she was CEO.

"This terrible tragedy did not happen on my watch ... All of my actions were to make Thomas Cook strong so that this did not happen again," said Green, who is still on gardening leave from the firm and "exploring options".

"I hold my head incredibly high about the work I did to save Thomas Cook, the work I did to put health and safety at the centre of the agenda ... My regret is that we didn't meet (the family) before the inquest and that I didn’t handle the inquest."

Green oversaw a programme of job cuts and store closures to help reduce Thomas Cook's debt and raise profits, sending its market valuation from 148 million pounds to around 2 billion pounds. Her exit in November rattled investors.

(Editing by Alison Williams)

By Neil Maidment