Grant was sacked Tuesday after he used vulgar language at the office of his club Geylang United on June 5, the Today newspaper reported.
"The vulgarity was targeted at the club ... it was unacceptable," Ong Yeok Phee, general manager of Geylang United was quoted as saying in the report.
Grant acknowledged he used strong language, but said he apologized for his behavior the following day in a letter to Ong, the report said.
According to the report, Grant said he swore because the club had only approved his leave application for 10 days, instead of the two weeks he had applied for.
"It is part of my culture and I did not say it to anyone's face," the Englishman was quoted as saying.
The city-state's football association told The Associated Press Wednesday it was currently investigating the matter.
Turkey has found itself in a circle of countries subject to US and European sanctions. Are they dangerous for Ankara? What is Turkey going to do in response?