Dr Robson also expects to be able to deliver England's blockbusting centre Manu Tuilagi to the selectors by Wednesday as the Lions' injury woes behind the scrum look to lessen in this vital week.
But it will be the improvement in North's injured hamstring that will be of most concern to Gatland over the next few days as he prepares to pick the best possible team to tackle the Qantas Wallabies.
"George is an exceptional character and is working very hard with Phil Pask. He is back to running today and he is doing very, very well. An accurate assessment on his availability for selection will take place on Wednesday," said Dr Robson.
"At the moment he has a great chance of being fit for this week. We wouldn't allow him to play if we felt he was at significant risk. Life is not without risk on any particular day, but we will only deliver back to the coaches for selection when we are confident medically he is fine."
While North pulled up before half-time in the midweek win over the Combined Country XV in Newcastle last week, Roberts injured his hamstring in the 47-17 victory over the HSBC Waratahs on Saturday.
According to Dr Robson it was an injury similar to the one that Lions lock Richie Gray suffered during the RBS 6 Nations, although nowhere near as severe.
"Jamie is very well this morning. His scan would indicate one thing, his demeanour would indicate another, but he is a gravc doubt for this week, there is no doubt about that," added Dr Robson.
"You only had to see how much discomfort he was in at the weekend, however, further assessment will be made later in the week. He has tweaked or strained his hamstring.
"It is not of any remote severity like Richie Gray, but was caused in a similar way. He got his leg extended and someone came down and flexed his back."
Tuilagi picked up his 'stinger' injury during the win over Queensland Reds and has been unavailable for selection since then. But he is back in training and has been working on the boxing pads to return the power to his shoulder.
"Manu is progressing on a daily basis and was back into training the other day. He will be assessed medically as we go through the week before we determine his availability, but he is doing exceptionally well," said Dr Robson.
While the experienced Bowe, who played in all three Tests in 2009, was able to get back to running this week, he is "highly unlikely" to be risked this weekend. He had an operation on his hand following the win over the Reds and was expected to be ruled out for at least two weeks.
Gatland had hoped he might creep into the selection equation this week, but Dr Robson has made it clear that the player will not be put at risk.
"Tommy Bowe is running today with a hand splint on. He is doing very well, and making remarkable progress, and we will get him reassessed by the hand surgeon later on this week," added Dr Robson.
"We will not allow Tommy to play unless we feel he is medically capable of playing. He had his small fracture stabalised almost a week ago and we are going to give him a couple of weeks to see how it he.
"He is perfectly comfortable and back running today. He has got his full grip strength back and the only problem we are going to have with him is reigning him back in a little.
"But while Tommy is highly unlikely to play this weekend, and Jamie is likely to miss out, everybody else is in line."