Tottenham manager Andre Villas-Boas related the news after his side's match against Fulham.
Winger Lennon was omitted from the Spurs squad that lost to Fulham after he suffered hamstring and groin injuries during Thursday night's defeat to Inter Milan.
Centre-back Dawson had to come off at half-time during the 1-0 defeat and was not expected to join up with the national team ahead of Friday's game against San Marino, which comes four days before their trip to Montenegro.
Lennon has featured prominently for Roy Hodgson this season after re-discovering his best form under Villas-Boas, but he will not be able to join up with the squad, the Portuguese said.
"Lennon won't be able to make it," Villas-Boas said.
"He has been struggling a lot with injuries. We risked him the other day against Inter Milan and he is on the brink of rupturing his muscle so we have decided to take him out of this game. He has groin and hamstring (injuries)."
Villas-Boas thinks Dawson has only a remote chance of joining up with the squad.
"He felt his (hamstring) muscle going in the first half.
"If we had brought him on in the second half we would be risking him a major injury.
"I think Michael will be assessed by the England team doctors, they will be in communication with our medical department to find out if he will make it or not. He probably won't, but it's something that they have to decide between them."
The news is a cruel blow for Dawson, who would have probably started Friday's game in San Marino.
The former Nottingham Forest defender was made available for transfer at the start of the season, but he has since forced his way back in to the Tottenham team with a series of fine displays.
The 29-year-old, who has four international caps, has not played for his country since the 2-0 win in Wales two years ago.