Cardinal Keith O'Brien, who described plans to create hybrid human-animal embryos for research as "monstrous", said he would be "only too happy" to attend such a meeting.
He also called for religious heads from other faiths to be involved, and urged the setting up of a body to advise the government on complex ethical issues.
At a public meeting in Kirkcaldy last night, Cardinal O'Brien said: "I see the way ahead through bringing together the churches and peoples of all faiths, along with scientists who are involved with the potential production of these human- animal hybrid embryos.
"I have been approached by MPs and asked by others in the media to consider meeting leading scientists who are currently involved in this area. I would be only too happy to agree to such a meeting and I am sure other church representatives and leaders of other faiths would also agree."
He added: "My only condition would be that the scientists were also willing to accept instruction from our Churches and peoples of faith on basic morality, on what human life really is, on the purpose of our life on earth and so on."
The Cardinal had claimed proposals in the Human Fertilisation and Embryology Bill could lead to experiments of "Frankenstein proportions".
Last Tuesday, amid mounting objections from senior Cabinet members and church leaders, Gordon Brown bowed to pressure for a free vote on crucial elements of the bill.
Supporters of the Bill argue that the use of hybrid embryos could lead to cures for diseases such as multiple sclerosis and Alzheimer's disease.
Scientists and politicians welcomed the cardinal's offer to discuss the issue, and Labour MP Jim Devine suggested a meeting could take place within weeks.
Professor Colin Blakemore, former head of the Medical Research Council, said he was "delighted" at the prospect of a meeting.
"I hope we can all enter into this dialogue with a shared willingness to listen to each other," he said.
"I hope the Church will accept that even scientists that do no profess religious beliefs do still have a strong moral compass - indeed it is exactly what drives many of us to search for treatments for incurable diseases."
Dr Stephen Minger, director of the stem cell biology laboratory at King's College London, said: "This is great news. We, as a scientific community, are keen to engage with all perspectives on this work to encourage the discussions to move forwards.
"We would like to see an ongoing dialogue that is based on an open exchange of ideas."
Cardinal O'Brien was speaking in the hall at St Bryce Kirk in Kirkcaldy, the church where Prime Minister Gordon Brown's father, a Church of Scotland minister, used to preach.
Mr Devine, who supports the Bill, had written to the Cardinal, calling on him to meet the scientists involved.
He said: "I'm delighted the Cardinal has decided to take this opportunity to meet scientists."
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