If Robson's assessment of Dyer's knee injury is correct, that would make him comfortably available for England's opening World Cup match against Sweden on 2 June.
Dyer's prospects received a boost when coach Sven-Goran Eriksson summoned the Newcastle midfielder to England's pre-tournament training camp in Dubai.
Our medical staff believe he is 10 days away from playing - that's playing, not training - and that wouldn't rule him out of England's plans
Newcastle manager Bobby Robson
Eriksson will give himself every opportunity to closely assess Dyer's fitness before Fifa's 21 May deadline for naming his squad.
But from a low point where he apparently needed "a miracle" following the challenge by Southampton defender Tahar El-Khalej, Dyer now has real hopes after making "excellent progress" according to Robson.
Robson said:"His knee looks fine.
"He has improved dramatically. Our medical staff believe he is 10 days away from playing - that's playing, not training - and that wouldn't rule him out of England's plans.
"He is swimming and jogging and his knee is holding firm. He is not in discomfort and he's making excellent progress.
"Now it's up to England to look at him and analyse the situation.
He will be coming out to join
us and we will see what happens then
"I spoke to Sven and Adam Crozier and passed all that information on to them.
"They are going to try to get hold of our physios so they can make their assessment.
"But our people are very optimistic and so is Kieron."
Dyer has not been assured of his World Cup place, but Eriksson will at least assess his fitness in Korea next Sunday and Monday before the deadline.
Eriksson technically has to make a decision on Dyer's fitness by next Tuesday night and will not want to take too great a risk given the midfielder's chequered injury history.
However, he could still name him in the squad and replace him if the problem persists under Fifa rules which allow injury-induced replacements to be made.
It's important that in the
next few days, our medical staff get the chance to assess him in detail and see
if he can prove his fitness
The news will be sure to give the squad a significant morale boost after Steven Gerrard joined Gary Neville in being ruled out of the tournament earlier this week.
Dyer will fly out to Dubai on Friday along with Football Association chief executive Adam Crozier and other FA figures.
He will then travel with the rest of the players to Korea on Saturday and be put through his paces over the next three days, while Trevor Sinclair remains with the squad on standby.
Eriksson declared: "This is very positive news. He will be coming out to join us and we will see what happens then.
"It seems that he can start running tomorrow and it's important that in the next few days, our medical staff get the chance to assess him in detail and see if he can prove his fitness."
However, in a statement on the FA's website, Eriksson insisted: "The important thing for any player at a World Cup is to be 100% fit."