The intruder, who is aged 18 and has a history of psychological problems, was arrested late on Saturday at the Paris monument after bursting past security and shouting "Allahu Akbar" ("God is Greatest").
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.
During questioning by police, the man stated he had wanted to carry out an attack against a soldier and was in contact with a member of Daesh terrorist group who encouraged him to act, the source said.
In January, a man attacked soldiers on guard near the Louvre museum in Paris with a machete, causing a slight injury to one of them.
Sources described the suspect as a French national born in Mauritania, western Africa, who left a psychiatric hospital in July.
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He nominated former Oil Minister Shahid Khaqan Abbasi as an interim prime minister. No civilian prime minister in Pakistan has ever completed a full term in office.
Monuments in the French capital are occasionally evacuated for security reasons as part of those measures.
The Eiffel Tower reopened to tourists as usual on Sunday morning.
He was wearing a Paris Saint-Germain football shirt.
The suspect was apparently also convicted a year ago for making death threats and "defending terrorism", for which he was given a four-month suspended sentence.
Paris mayor Anne Hidalgo praised the "professionalism and reactions" of security forces at the site.