The SNP had promised to publish the draft legislation in their first 100 days in government after the Scottish election in May.
Swinney said: “The world has changed fundamentally since 2014. So has the UK, and so has Scotland.
“Scotland is ready. The Scottish people are ready. It is time the UK government delivered.”
The first minister said he would have “more to say about Scotland’s constitutional future in the coming weeks”.
He added: “Fundamentally, nobody will take better decisions about Scotland’s future than the people who live here.
“That future belongs to the people of Scotland – and no-one else.”
In line with current legislation, registered voters over the age of 16, as well as qualifying foreign nationals, would be able to vote in a second referendum.
