The veteran prop was expected to miss the entire tournament after almost severing his thumb while chopping wood for a barbecue. However, Murray, who has also overcome a hamstring problem, made a surprise return for his club, Worcester, in their English Premiership match against Sale Sharks last weekend.
He is likely to be in the starting XV when Worcester face Harlequins at the Twickenham Stoop on Saturday and could figure in Scotland coach Scott Johnson's plans for France's visit to Murrayfield seven days later.
The director of rugby at Worcester, Dean Ryan - who coached the Scots pack in last season's championship -said: "The news about Euan is positive. The healing time for his injury was pretty good.
"He obviously suffered a significant cut and there were some doubts about the ability of his body to grow back the skin. We were without Euan for more than three months and our lack of tighthead cover made things difficult for us."
Scotland also had worries in the No.3 berth during the first half of last Saturday's win over Italy in Rome. Glasgow's Moray Low was substituted after only 36 minutes after the referee, Steve Walsh, awarded a series of scrummage penalties against him.
Despite Murray's lack of game time, Johnson will be tempted to throw him straight into the action against France, or could use him as an impact substitute in the final quarter. Johnson is due to name his team on next Tuesday.
Meanwhile, Johnny Sexton, the Ireland fly-half, also has a thumb injury but Joe Schmidt, the head couch, expects him to be fit to face Italy in Dublin on March 8.
Sexton has his hand in a splint but could nevertheless turn out for Racing Metro against Castres in the French Top 14 this weekend.
Schmidt said: "He can still pass and do everything but we're going to get some more feedback over the next few days. Hopefully, he'll be fine.
"He's gone back to Paris and we have some scans to be read but I predict he should be available for the final two games."