A judicial official says a man holding a knife said "Allahu akbar" - the Arabic phrase for "God is great" - and tried to access a secure zone.
The incident was initially treated as a criminal matter, but was later designated a counter-terrorism case because of the suspect's statements to police and his apparent radicalisation, the source said.
Under questioning, the man said he wanted to attack a soldier and that a member of Islamic State had encouraged him to act, a source close to the investigation said.
In February, a man armed with a machete attacked four soldiers on patrol at Paris's Louvre Museum, while in April another extremist shot and killed a policeman on the Champs Elysees.
But the suspect, a French national born in Mauritania in 1998, is thought to have psychiatric problems.
Last night, the Eiffel Tower was lit in red and blue in the colours of the football club "Paris Saint Germain" in honour of the arrival of the Brazilian star Neymar.
France has been on high alert since a series of terrorist attacks in 2015 and 2016 in which more than 230 people died.
The tower reopened this morning as usual. The wave of terrorist attacks of the last two years has so far left a total of 239 dead and in recent months the attacks have been directed in particular against security forces.