Although the German was penalised during the race and lost further licence points, it was speculated that FIA president Jean Todt wanted to pursue the matter further.
Amid suggestions that a race ban could be applied, it is suggested Todt is unhappy that Vettel re-offended despite being sternly warned after Mexico last year about his radio tirade against Charlie Whiting.
The FIA confirmed that on Monday, July 3, it will "further examine the causes of the incident in order to evaluate whether further action is necessary".
The outcome will then be announced before next weekend's Austrian Grand Prix, F1's governing body added.
Monday, July 3 is also the Ferrari driver's 30th birthday.
Mercedes team chairman and F1 legend Niki Lauda told newspaper Osterreich that Vettel's decision to drive into Hamilton in Baku was "unbelievable".
"What makes it even worse is that Vettel took absolutely no blame - although that is always so with him," he said.
"I can understand Lewis's anger and I'm really angry as well. Why couldn't he just say 'It was my mistake, I screwed up'?
Lauda also seemed to agree that the FIA is right to re-open the matter, saying that the penalty given by the stewards in Azerbaijan was "a joke".