The intended mother, also British, is said to have taken the healthy boy, but refused to accept his twin sister because she has a severe muscular condition called Congenital Myotonic Dystrophy.
The surrogate mother, known only as Jenny, told The Sun: "I'll never forget what she said on the phone.
"I remember her saying to me, 'She'd be a f****** dribbling cabbage! Who would want to adopt her? No one would want to adopt a disabled child'.
"I was shocked."
Jenny and her partner Mark signed a surrogacy deal with the intended parents in return for £12,000 expenses.
Two embryos were implanted into Jenny's womb and a few weeks later she learnt she was carrying twins.
The girl's disability was diagnosed after Jenny had given birth.
"I got into surrogacy because I wanted to help a mother who couldn't have children," she said.
"But I feel very angry because of what they've done to her."
The couple, David and Wendy Farnell, dispute surrogate Pattharamon Janbua's account.
Jenny said: "We are living proof what happened with Gammy in Thailand can happen here in Britain."
Mark added: "How could we possibly sign over to somebody showing a disregard of the child's health?"
The two couples subsequently attended mediation meetings with the Children and Family Court Advisory and Support Service, which gives an independent assessment to help judges hearing child custody and care cases.
These hearings are confidential, but The Sun reported that the intended parents expressed concerns about the girl's disability and their desire to take her on.
The couples agreed Jenny and Mark would keep the girl and the boy would stay with the intended parents.
"We did not set out to have another child so it was a big decision to make," said Jenny.
"We just wanted to help another family. That's why we got into this."