Data from state statistics institute INSEE released this week showed growth in charges slowed in January when lenders usually set their rates for the year, after they raised fees in 2017 at the highest pace in nearly 20 years.
The INSEE data showed that consumer price index of banking charges rose by 1.05 percent over one year in January, down from 2.49 percent in January of last year.
"Banking services will not be free. The security of a deposit is by essence costly," Frederic Oudea, chief executive of Societe Generale said earlier this year.
Established banks, such as BNP Paribas, Societe Generale and Credit Agricole are facing increased competition from the likes of new online low-cost banks such as N26 or Orange bank.
Consultancy firm EY estimated that the new low-cost online banks now have a five percent share of the French market which totals 120 million accounts.
Analysts had predicted that this could spark a price war as happened in France's telecoms sector, while the established banks have invested a lot to retain clients with their own low-cost offers, such as BNP Paribas' Hello Bank!, Boursorama for SocGen or EKO for cooperative group Credit Agricole.
"We are very far from the situation which was observed in the French telecom industry in 2012 when telecom operators had to face a period of several months of declines in average revenue per user," said Axel Duplan, a senior manager at EY.
That is partly because French retail banking clients have so far tended not to leave their established lender, instead open new accounts at lower cost online banks for daily use.
"Customers don't yet change banks frequently, not seeing them as easily replaceable as digital providers, but we believe this shift could occur in the medium term as the new legal framework eases banking mobility," analysts at ratings agency S&P said in a note.
French bank clients like not only having an account but also savings products, life, auto, property, and health insurance that online banks are so far not able to provide, S&P said.
BNP Paribas' chief financial officer Lars Machenil said in February it was yet to see the impact of the arrival of Orange bank and a step-up in competition.
To view a graphic on French banking fees, click: http://reut.rs/2EiNHsi
(Reporting by Maya Nikolaeva; Graphic by Leigh Thomas; Editing by Alexander Smith)
By Maya Nikolaeva