The Bayern Munich striker was forced off with an unspecified problem in his side's Bundesliga draw with Eintracht Frankfurt at the weekend, having just recovered from a previous scare.
Bayern boss Vincent Kompany could not say at the time how severe Kane's issue was, but the former Tottenham Hotspur man was cleared to join up with the England squad, where he underwent further assessment.
Kane has remained at St George's Park after those tests, but his chances of starting in Thursday's showdown with Greece appear slim following the latest update from the England camp.
The Three Lions confirmed that Kane was still following an individual training programme ahead of the clash with the 2004 European champions, whom he is therefore unlikely to start against.
England suffer new injury scare alongside Kane concern
With Kane potentially only making the bench at Wembley, either Ollie Watkins or Dominic Solanke - called up for the first time in seven years this month - should spearhead the attack for Lee Carsley's side, as Kane is kept in reserve before the trip to Finland on Sunday.
The 31-year-old is not the only injury concern for interim manager Carsley to contend with, as England also announced that Manchester City attacker Jack Grealish missed training as a precautionary measure due to a knock.
The 29-year-old quickly got into Carsley's good books last month, starting and scoring in the 2-0 win over the Republic of Ireland - whom he represented as a youth player - before also playing the full 90 against Finland.
Grealish's possible absence could lead to a three-way battle between Cole Palmer, Jude Bellingham and Phil Foden for the number 10 spot in the England team, although Bellingham could alternatively start deeper alongside Declan Rice.
England have already lost Kobbie Mainoo, Morgan Gibbs-White and Ezri Konsa to injuries since naming their squad, and Carsley initially did not select any replacements, but he has now reinforced his squad with two new arrivals.
Curtis Jones, Tino Livramento called up to England squad
Liverpool midfielder Curtis Jones and Newcastle United right-back Tino Livramento have been added to the Three Lions' squad, and both will be pining for their first senior caps this month.
Jones - part of Carsley's triumphant Under-21 Euros troupe last year - was called up to Gareth Southgate's pre-Euro 2024 squad in June before being cut from the final selection, but he was unavailable for September's games due to a muscle injury.
Meanwhile, Livramento was selected for Carsley's first two matches in charge, but the 21-year-old failed to make it off the bench in both games against Ireland and Finland.
England sit second in League B Group 2 of the Nations League, only behind Greece on goal difference, but they will go three points clear of the Pirate Ship at the summit with a third straight win under Carsley.