Burns underwent surgery to have a titanium plate inserted after Raymundo Beltran inflicted a clean break of the Scot's jaw early in their world title bout on September 7.
Less than four weeks after his injury, he reported progress in his recovery via his official Twitter account. Burns wrote: "Was at hospital last week and jaw's getting there, can resume light training but still no boxing for a few weeks. Going nuts here! Haha."
His promoter Eddie Hearn cast doubt on the 30-year-old's future in the sport in the aftermath of the operation but Alex Morrison, the boxer's manager, insisted his charge would fight on and hoped to give a rematch to Beltran.
Burns retained his belt after the judges declared the Glasgow fight a draw, much to the disgust of the Mexican and his camp.
Meanwhile, Amir Khan's team has dismissed reports that the Bolton fighter has pulled out of his proposed December fight with Devon Alexander in order to face Floyd Mayweather.
Khan is lined up to meet Alexander for the International Boxing Federation welterweight world title on December 7 with only the venue to be decided, but reports on Tuesday night suggested that Khan was to scrap that bout in favour of a more lucrative match-up against the undefeated Mayweather.
His team moved swiftly to dismiss them, insisting that the 26-year-old is fully focused on taking on Alexander before the end of the year. "Team Khan would like to dispel recent reports that Amir Khan has pulled out of a potential match-up against Devon Alexander on December 7, taking up a fight against Floyd Mayweather in May of next year instead," read a statement. "Contrary to this story, Amir Khan is currently in training camp in San Francisco with the aim of fighting again in December against Alexander - once an agreement can be reached."
Khan has been rebuilding his career following a miserable couple of years in which he lost his light-welterweight belts in defeats by Lamont Peterson and Danny Garcia.