Scottish Secretary David Mundell confirmed the move to the Commons Scottish Affairs committee saying he could see no reason for it not to be devolved.
It is thought abortion will be added to the Scotland Bill that will see a draft of new powers head to Edinburgh following the independence referendum.
Abortion is already devolved to Northern Ireland.
The SNP government has already said it would not seek to change the law.
First Minister Nicola Sturgeon told the Scottish Parliament in September that her government would write to women's groups to confirm no change in the law, should it be devolved.
But one of Scotland's leading Christian charities is calling for the issue to be debated now.
CARE for Scotland said MSPs should take advantage of the opportunity to debate the issue for the first time in the Scottish Parliament.
National Director Dr Stuart Weir said: "We welcome the fact abortion is to be devolved to the Scottish Parliament.
"Only recently MSPs debated a crucial end of life issue when the Parliament rejected assisted suicide by an overwhelming majority.
"It makes plain constitutional sense therefore for this new power to be given to Holyrood so the Parliament here has power over start of life issues as well.
"The transfer of such a major new power clearly merits a full Parliamentary debate and MSPs should take advantage of this opportunity as soon as possible.
"In recent years the Scottish Parliament has debated assisted suicide and other complicated topics and proved itself equal to the task so it makes obvious sense to debate abortion laws in Scotland following this transfer."
David Mundell MP said: "I understand that abortion law and whether it should be controlled by Holyrood are matters that many people feel very strongly about. I respect that.
"The Government has reflected very carefully on the points that have been made and I can today inform the Committee that we will bring forward an amendment to the Scotland Bill so that abortion law can be devolved to the Scottish Parliament.
"Holyrood already has responsibility for dealing with end of life issues. It has responsibility for the NHS and for criminal justice in Scotland.
"I do not see a convincing constitutional reason for why abortion law should not be devolved and that is what has led me to this decision."