At 1445 GMT, shares in SAS -- in which Denmark, Norway and Sweden together own 50 percent -- were up more than 10 percent on a Stockholm stock exchange up 1.83 percent.
When the market closed at 1630 GMT, SAS shares had gained 11.11 percent to 25 kronor ($3.70) on a market up 1.84 percent.
The gains came the first trading day after a report in Danish business daily Boersen said three major European airlines -- Air France-KLM, Lufthansa and British Airways would be ready to enter a "bidding war" to buy SAS.
The newspaper quoted an unnamed source as saying "when the process starts, I think all the major aviation players will be interested," referring to when the Scandinavian governments will be ready to sell their stake in the airline.
The Swedish government is the single largest shareholder in SAS with a 21.4-percent-stake, while Denmark and Norway each hold 14.3 percent. The rest is privately owned.
In February 2010, the Swedish government said it was interested in reducing its holding in SAS at an "opportune moment" and Denmark and Norway followed suit.
SAS has been hard-hit by the rise of low-cost airline Norwegian and by plunging passenger traffic numbers in the wake of the global economic crisis.
Rumours of a sell-off have been circulating since 2008, with Lufthansa seen as the most probable buyer.
Late last month, SAS shares soared on rumours that Lufthansa, already an SAS partner on some routes, could make an offer for the Scandinavian company as early as the first quarter of 2011.
An SAS spokesman reached Monday said the company would not comment on market rumours.