Christi and Bobby Shepherd were poisoned by carbon monoxide at the Louis Corcyra Beach Hotel in 2006.
Their parents, Sharon Wood and Neil Shepherd, met the tour operator's chief executive Peter Fankhauser earlier.
Speaking at a press conference, Mrs Wood said she accepted it "may be time to look to the future".
The amount of money being given to the family was not disclosed, but Mr Shepherd said they planned to make donations to a series of charities.
Mrs Wood called upon Thomas Cook "to push forward our request" to have the bungalow where the children died be demolished and turned into a playground in their memory.
"Nothing can give us back our children and the carefree lives we once led," Mrs Wood said.
"I hope Thomas Cook and everyone who defended its wrong-doing now realise the impact on families like ours and they will learn the lessons they need to learn from this tragedy."
Christi and Bobby, from Horbury, near Wakefield, were on holiday with Mr Shepherd and his now wife, Ruth, when they died.
It was later discovered they were overcome by fumes from a faulty boiler.
A criminal trial held in Greece in 2010 cleared Thomas Cook of any responsibility and awarded the firm damages against the hotel's owner.
But an inquest jury ruled last week the children had been unlawfully killed and said the tour operator had "breached its duty of care".
Mr Shepherd and Mrs Wood criticised the firm for not apologising to them directly during the two-week hearing.
Mr Fankhauser made a public apology on Wednesday and admitted the firm "could have done better in the past nine years" in the way they had responded to the tragedy.
It later emerged two hotel staff who had been convicted over the children's deaths were found working at hotels where Thomas Cook sent holidaymakers.
Mr Fankhauser said he felt "physically sick" when he discovered the staff were working at the hotels and  one of the employees had subsequently been sacked from his post in Crete.
The travel firm said it had stopped sending holidaymakers to another Greek hotel where the other was employed.
The company has donated half its Â£3m payout from its insurers to children's charity Unicef.
The family of two children killed on a Thomas Cook holiday in Corfu will receive a "financial gesture of goodwill" from the firm.