Fillon, chosen in November as the Republican party candidate for the presidency, has previously rejected the claims, declaring he has "nothing to hide."
It is alleged that Fillon's wife, Penelope, and children have received public funds for work they did not do.
In a statement, the financial prosecutor's office said it had received the results Wednesday of a preliminary investigation into the allegations of embezzlement of public funds.
"Many elements are already collected and the case cannot be dropped at this moment," the statement said.
"Investigations will continue in strict compliance with the rules governing the Code of Criminal Procedure. The sole mission of the national financial prosecutor is to apply the law."
Lawyers for Fillon responded that the investigation into their client "violates the rules" set down for the financial prosecutor.
"This official statement does not mention the fact that after three weeks of investigation, there are not enough elements to require further investigation," lawyers Antonin Lévy and Pierre Cornut-Gentille said in a statement.
"We still think that the investigation violates the rules of the National Financial Prosecutor and the principle of the division of powers, which is even more serious. The investigation must be carried out in a legal, serene way.
"This is important as the first round of the presidential election is taking place in a few weeks."
The 62-year-old conservative candidate has come under increasing pressure since the investigative newspaper Le Canard Enchainé first alleged that his wife and two of his adult children were paid hundreds of thousands of euros for work they did not do.
Earlier this month, Fillon told a news conference that he had been attacked with "incredible violence which has never been seen in the Fifth Republic."
The couple has been questioned by police. Both have denied the allegations and no one has been charged.
Levy, the Fillons' lawyer, told CNN earlier this month that "the fact that Penelope Fillon's work is real was proven to the investigators."
The accusations have dealt a huge a blow to Fillon, who was already facing a fragmented political field in a bitter presidential campaign.
He faces stiff competition from Marine Le Pen
of the far-right National Front and independent candidate Emmanuel Macron. On the left, Benoit Hamon clinched the socialist nomination last month over former Prime Minister Manuel Valls.
The first round of the election will be held on April 23, with a runoff on May 7 if no candidate gets more than 50% of the vote.