The two-time world champion, who joined from Ferrari at the end of last season, will miss Sunday's Australian Grand Prix on doctor's orders after spending three nights in hospital following the crash in Barcelona.
Alonso's condition has been the subject of media speculation after a Spanish newspaper reported the 33-year-old could not remember he was a world champion when doctors tested his recall after he regained consciousness.
"We are working with McLaren and our medical department is still looking at all the information we can find," FIA race director Charlie Whiting told reporters on Thursday.
"We are still talking to McLaren about anything mechanical. Our medical department is talking to people who treated him in hospital and that is really all I can say on that."
Last month, McLaren blamed gusting winds for the crash and firmly denied that mechanical failure had played a part. The team conceded he had suffered a temporary loss of memory but had recovered.
Alonso also joked about his condition on Twitter.
McLaren have said Alonso will be back in the car for the Malaysian Grand Prix, the second round of the season after Melbourne.
Dane Kevin Magnussen will drive in Alonso's place at Albert Park, partnering Briton Jenson Button.
McLaren, who are starting up a new partnership with Honda, have done the fewest laps of any of the top teams in testing and been repeatedly sidelined by problems with the car.
(Editing by Sudipto Ganguly)
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