The "clear and unequivocal" apology on behalf of the Glasgow-based company came in a meeting with business secretary Vince Cable last week. Mr Brodie's resignation was not accepted.
The Government agency has been accused of using "Wonga-style" tactics by sending out letters under the name of Smith Lawson and Company Recovery Services - a firm it set up - demanding repayment from graduates who were in arrears.
In a ministerial statement, universities minister David Willetts said ministers believed Mr Brodie had a "very important" job to do at the SLC, and it would be unfair for him to take the blame for a practice that was nearly 10 years old.
More than 300,000 graduates received letters from Smith Lawson and Company over a period of 10 years.
Letters were sent to more than 300,000 graduates but the system was stopped in June after the Financial Conduct Authority's decision to sanction payday lender Wonga for "aggressive and misleading practices".