Life, an anti-abortion group, said the public had donated more than pounds 10,000 to help the woman and others who may be faced with the same dilemma.
In what is believed to be the first "selective termination" of its kind in Britain, Phillip Bennett, professor of obstetrics and gynaecology at Queen Charlotte's Hospital, London, has agreed to abort one foetus because the mother has said she could not cope with two babies.
The patient, a 28-year-old single mother who has one child already, is said to be in "straitened circumstances".
Selective termination is usually used when in vitro fertilisation results in a multiple pregnancies.
Doctors can choose to abort one of the foetuses if it shows a sign of abnormality or if a number of embryos implanted in the womb increase the risk of complications.
Life said it was offered pounds 1,000 on Sunday by an anonymous donor to persuade the woman to cancel the operation and yesterday another donor offered pounds 5,000 to go into a national fund to help her and people in a similar situation. Life will also offer the woman accommodation before and after the birth.
Following news of the pounds 1,000 offer, an Italian pro-life organisation has also offered to give the woman financial support. Josephine Quintavalle said Movement for Life would be able to offer her pounds 140 a month if she chose not to have the abortion, as part of its Project Gemma sponsoring scheme.
Life's chairman, Professor John Scarisbrick, said:"We have sent a message to Professor Bennett asking him to tell her that.
"We would not want her to think we are trying to bribe her," he added. "We are trying to say: 'Please don't do something which is horrible, which is liable to haunt you for ever, and which is going to cause all sorts of trauma and grief'."
It is not clear whether the woman knows of the cash offers or whether she has already undergone the termination. Last night Hammersmith Hospitals NHS Trust declined to comment on the case. Its spokesman, Jonathan Street, said: "The professor is not prepared to talk to anyone about it and the hospital has no reason to. It is a matter of clinical judgment and we are quite intent on preserving the confidentiality of this woman."
Sir David Steel MP, who put forward the 1967 Abortion Act, said the case was "bringing forth the usual crop of statements from people opposed to all abortion who are therefore ready to use this highly unusual and marginal case to advance their campaign to abolish legal abortion in Britain.
"This would merely return the country to the dark days of illegal abortion, which would apparently salve the consciences of the anti-abortion organisations. As I have said many times, the law requires two doctors to act in good faith in sanctioning or refusing an abortion. Presumably they will do so in this case, about which they know the details and have discussed with the parents while the rest of us know little."