The 49-year-old Sonko, who came third in the 2019 presidential election, has been at the centre of a bitter stand-off with the state that has lasted more than two years and sparked often deadly unrest.
Sonko's lawyer, Cire Cledor Ly, said the candidacy had been rejected on the grounds that the application was incomplete.
"When we entered, (Council) President Badio Camara immediately notified us that (Sonko's) file was incomplete," he said.
More than 90 candidates have put their names forward to the Constitutional Council, which is due to announce the list of presidential contenders on January 20.
President Macky Sall in July announced that he would not seek a third term in the February 25 poll, handpicking his prime minister, Amadou Ba, as his coalition's presidential candidate.
Sonko had in December filed his candidacy with the Constitutional Council, despite the state's refusal to provide him the necessary documents to run.
They argued that Sonko had been removed from the electoral register after being sentenced in June to two years' imprisonment for morally corrupting a young person.
Sonko's lawyers had said they would file his presidential candidacy anyway.
On Friday, Ly said the Constitutional Council president told him that "the files, the accompanying letters and the attached documents were received and checked by the commission which concluded that one document was missing and that the candidacy file was incomplete."
'Electoral farce'
Sonko's lawyer denounced the Council's decision as an "electoral farce" and suggested he would lodge "the appeals provided for by law."
"The commission's composition was irregular because the law stipulates that this verification must be carried out in the presence of the candidate or the proxy," he said.
"There is a desire to move towards elections which from the outset lack transparency and which in any case will not reflect the will of the nation."
Sonko's party television said the file was incomplete because it lacked a certificate from the CDC deposit bank, where a cheque for 30 million CFA francs (nearly $50,000) must be deposited in order to run in the presidential election.
A day earlier, the opposition figure's chances of running for president had been thrown into jeopardy after Senegal's Supreme Court upheld a six-month suspended sentence handed to him for defamation.
Sonko's camp had maintained he still had the right to run in the election since a judge in December ordered that he be reinstated on the electoral roll.
The firebrand figurehead has been jailed since the end of July on other charges, including calling for insurrection, conspiracy with terrorist groups and endangering state security.
He has generated a passionate following among Senegal's disaffected youth, striking a chord with his pan-Africanist rhetoric and tough stance on former colonial power France.
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