Both Mr Murphy and the First Minister are due to speak at a conference this week.
And now Mr Murphy has written to the SNP leader, suggesting the first ever public debate between a Scottish Secretary and First Minister.
In his letter the Scottish Secretary told Mr Salmond that the conference - Scotland: The Possible Future - presented a “good opportunity” for them to debate the issues facing Scotland.
Mr Murphy also claimed there were many questions that “remain unanswered” about the SNP`s flagship policy of independence, adding that the event could be a “fitting forum” to discuss this.
Both Mr Murphy and Mr Salmond are keynote speakers at the conference, which is taking place in Edinburgh on Wednesday.
In his letter Mr Murphy points out to the First Minister that their speeches are “separated only by a coffee break”.
He suggested: “I thought it might be more interesting for the audience if, instead of speaking back to back, we went head to head in a straight debate on the future of Scotland.”
The UK Government minister continued: “Many questions remain unanswered on the detail of your proposal of separating Scotland from the United Kingdom and this would be a fitting forum for this discussion.
“I am certain the organisers would allow us to combine our allotted slots and have a debate rather than just two separate speeches.”
Mr Murphy added: “I am sure you will agree with me that the public exchange of views is an essential and accessible part of any democracy.
“We have very different views on what Scotland’s future should look like, and the event presents a good - and long overdue - opportunity for a debate between us to finally take place.
“I accept it is sometimes difficult to change our diaries at this short notice. However, we are both due to be in the same place at the same time and I think we risk missing a good opportunity to talk directly about some important issues facing the people of Scotland. I hope you agree.”
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