By Lefteris Papadimas

ATHENS (Reuters) - Greek banks must replace almost a third of their board members by September in a bailout-mandated drive to strengthen corporate governance, bankers with knowledge of the matter told Reuters.

Greek banks traditionally have businessmen, union leaders and in some cases politicians on their boards. But under its third international bailout, Greece agreed to try to 'de-politicise' links between government and the banks, boost board-level expertise and improve corporate governance, one banker said.

The country's HFSF bailout fund hired external consultants to review the boards and they have said that up to 18 directors on the boards of four different banks must be replaced.

"They (Banks) should replace about 15 to 18 board members, mainly non executive, several businessmen and in some cases union (members)", a senior banker with knowledge of the issue told Reuters without providing details. The first banker confirmed the number.

Greece's big four banks - National Bank <NBGr.AT>, Piraeus Bank <BOPr.AT>, Alpha Bank <ACBr.AT> Eurobank <EURBr.AT> - have 58 directors in total.

HFSF confirmed that it had sent the consultants' report to the central bank and the banks but did not give details.

HFSF has stakes ranging from 40 to 2.4 percent in the banks which were bailed out three times during Greece's debt crisis. The latest was in late 2015 when they received 5.4 billion euros($5.98 billion) from the European Stability Mechanism channeled through HFSF.

Under the bailout rules, directors must have at least 10 years of banking experience at senior managerial level, have no prominent position in a political movement or to have held a position in government in the last four years.

"There will be a few changes in top bankers," the senior banker said.

Louka Katseli, president of National Bank (NBG), Greece's second largest bank by market capitalization, could possibly leave, the banker said.

Katseli is a former economy minister who served in government in 2009. She was also head of a small party known as Social Pact until early 2015.

She supported the ruling leftist party in 2015 elections and was made president of National Bank's board when Greek Prime Minister Alexis Tsipras won the vote.

But the Greek government is not happy to let her go, another banking source said.

"The government is in discussions with the country's lenders to change the specific term (political position) and keep Katseli as president of National Bank", the source said.

National Bank declined to comment.

Consulting firm Spencer Stuart carried out the review in April and it was sent to the Central Bank of Greece in the past week.

"Each of the four systemic banks has received the part of the report which concerns it," the first banker said.

"The banks have until the end of July to inform the HFSF of the changes in their boards and implement them by September at the latest", he said.

Eurobank and Piraeus confirmed that they had received the evaluation report, but did not want to comment further. Alpha Bank was not immediately available for comment.

(Reporting Lefteris Papadimas. Editing by Jane Merriman)