The investigation into breach of confidence and fraud follows a preliminary inquiry started in June.
It means that the case will be taken over by an examining magistrate, who will be able to summon witnesses with a view to possibly laying charges.
All three of the party's ministers in Macron's first government - Bayrou, European affairs minister Marielle de Sarnez and defence minister Sylvie Goulard - resigned ahead of a cabinet reshuffle when the preliminary inquiry was opened.
The allegations were first made by National Front MEP Sophie Montel, who along with several other members of her party, including its leader Marine Le Pen, was already the subject of an investigation.
Since then a former party worker, Matthieu Lamarre, has come forward to say that he was paid by the European parliament while working for the party in France.
Bayrou, while denying any wrongdoing, has said that some employees were sent to the European parliament when the party, which he founded in 2007, was hit by financial difficulties.
Montel has also accused a number of other French politicians of using European parliament funds for party purposes, most recently hard-left leader Jean-Luc Mélenchon.