The Scottish Conservatives lodged the motion on Monday afternoon and hope to force a vote on the issue on Tuesday afternoon unless the legal advice is published by the Scottish Government.
The move comes after the party gave Mr Swinney a deadline of 24 hours to publish the legal advice or face his second no-confidence vote in six months.
It is hoped the vote can take place before Nicola Sturgeon’s appearance in front of the Holyrood inquiry into the handling harassment complaints against Mr Salmond on Wednesday.
Holyrood has twice voted in favour of the Scottish Government releasing the advice, which is seen as key to understanding why the judicial review was not conceded earlier.
Its eventual concession in January 2019 led to costs of more than £500,000 of taxpayers’ money.
Scottish Conservative leader Douglas Ross said the public “deserves to know what happened” and called on other opposition parties to back the move.
He said: “Women were let down and more than £500,000 of taxpayers’ money was lost.
“The public deserves to know what happened and the legal advice is crucial to uncovering what mistakes were made.
“We gave John Swinney one more opportunity to respect the will of the Scottish Parliament. He has failed to do so.
“Opposition parties have united twice already to demand the legal advice is released. We now call on them to do so a third time.
“We are still willing to withdraw this motion at any point, if the government respects the will of the Scottish Parliament.”
The motion lodged by the party states: “That the Parliament has no confidence in the deputy first minister, in light of the Scottish Government's continued failure to publish legal advice called for in two resolutions of the Parliament on 4 and 25 November 2020.”
The Scottish Liberal Democrats said on Sunday they would back the motion, with the Scottish Greens likely to have the casting vote.
The no-confidence vote could see Mr Swinney lose his position as deputy first minister and education secretary and comes after he narrowly survived a similar motion in August.