An agreement could be signed by the end of this week and CMA CGM would receive payment at the end of November, French financial daily Les Echos reported.
Reports of an imminent deal follow the collapse of negotiations with several potential bidders over the past six months, including Colony Capital, a U.S. investment group, Belgian billionaire Albert Frere and Qatar Holding.
The French government's strategic investment fund, FSI, is expected to invest in CMA CGM after the carrier reaches agreement with Yildirim. CMA CGM declined to comment on the reports.
Yildirim has interests in coal and steel trading, logistics, cargo handling and shipping services. Its subsidiaries include Yilport Container Terminal and Port Operations and Yilmar Shipping.
CMA CGM's search for outside investment to bolster its balance sheet following the slump in the container shipping market has been flanked by negotiations with creditor banks to restructure around $5.6 billion of debt taken on to expand its fleet, the world's third largest.
Pressure on the company has eased after it swung to a first half net profit of $864 million from a year earlier loss of $581 million as revenue surged 41 percent to $6.8 billion on higher traffic and freight rates.