It came as an announcement is due in the next two weeks over the chosen route, which The Scotsman understands will be one of the shorter of the 11 options.
However, Mr Matheson told the Scottish Parliament’s connectivity committee he would be “hesitant” to say how long the project would take because there were factors outwith his control, such as statutory consultations.
Campaigners want the current route upgraded rather than bypassed, and completed in three years, by May 2024.
The A83 has been frequently blocked by landslides, some of which have also closed a parallel diversion route, the Old Military Road, below it.
The road is the main route between Glasgow and the Kintyre peninsula and other parts of Argyll, with blockages forcing a detour of up to around 60 miles.
Some £15 million has been spent so far on mitigation measures such as fencing and pits to catch debris, but their effectiveness has been hampered by the random locations of the landslips.
The 11 bypass options include upgrading the current route, a new road through Glen Kinglas to the north, and one through Glen Fyne, slightly further north still.
The shortlist also includes bridges or tunnels across the Clyde via the Cowal peninsula or Bute, and across Loch Long and Loch Fyne.
No estimated costs or timescales for the options have been published.
A transport source told The Scotsman: “The argument was between something that could be done (relatively) quickly and a more ambitious project, such as fixed links across the Clyde).
"I expect quick to win.”
The source said a land route was most likely – which would appear to chime with Mr Matheson underlining the importance of completing the scheme quickly.
Mr Matheson said: “I hope by the course of the next couple of weeks before the recess to set out the preferred corridor that has been identified through the consultation exercise.
"That will then draw down to the specific options within that particular corridor for delivering a permanent solution to the challenges that we have on the A83 at the Rest and Be Thankful.
"I have also asked officials in Transport Scotland to consider how they can compress the time frame for the completion of the project as much as possible.
"I have made it very clear to my officials that they should be looking at this as, in effect, an emergency project – one that requires to be taken forward at rapid speed.
"I’ve asked them to have a mindset that looks at it as an emergency as opposed to being a normal major infrastructure project.”
John Gurr, chair of The Rest and Be Thankful Campaign; said: “Whilst this announcement is good news, we fear without a mention of a completion date it is simply lip service, especially so near to the elections.
“The Scottish Government has been “hesitant” about fixing this for the past eight years.
"We would like to see some realistic timescales confirmed by Michael Matheson
"This crisis has cut off Argyll, and caused delays and additional costs for business, and we need Michael Matheson to use emergency powers to push through a solution.”