Best, who has been ruled out of the remainder of the World Cup, was rushed to Bordeaux's Pellegrin Hospital on Wednesday night after developing a loss of sensation down his right side.
The 29-year-old Ulster prop also complained of a mild headache and a difficulty in speaking and underwent a series of tests, including an MRI scan, an angiogram and a CT.
The tests were repeated on Thursday morning and Best was then transferred to the nearby Haut Leveque - one of France's leading cardiological hospitals - for further investigation.
As yet doctors have been unable to make a clear diagnosis on Best's condition and Ireland coach Eddie O'Sullivan this afternoon refused to speculate on the nature of the condition.
O'Sullivan said: "We are concerned. There is a certain amount of concern because there isn't a diagnosis at this stage and I suppose that's the worrying thing.
"I wouldn't speculate on what Simon's condition is and I don't think anyone should speculate about it at this point. It would be wrong for everybody, including his family.
"We don't have a diagnosis and hopefully it's nothing like that (a stroke) - hopefully it might be something very, very small.
"In the meantime we have to keep our fingers crossed."
Munster prop Tony Buckley has been called up to replace Best, who was named on the bench for Sunday's World Cup clash with Argentina in Paris.
Buckley will depart for France as soon as possible with Best's Ulster team-mate Bryan Young filling the vacancy on the bench against the Pumas.
A decision on who will start at hooker in the Pool D finale will be made later on Thursday with Simon's brother Rory Best (dislocated thumb) and Jerry Flannery (shoulder stinger) undergoing fitness tests.
Simon's condition may well affect O'Sullivan's decision on who wears the number two jersey.
O'Sullivan said: "Rory's a bit upset with the news but he knows that Simon is in the best possible hands and that everything has been done for him.
"At this stage it's just a case of waiting to find what the problem is. I spoke to Rory at lunch and he was ready to train this afternoon."