The chairman of the Student Loans Company (SLC), Christian Brodie, has apologised to ministers and offered to resign over "misleading" letters sent out by the agency in an attempt to recover debts, it has been revealed.

The "clear and unequivocal" apology on behalf of the SLC came in a meeting with the business secretary, Vince Cable, last week. Brodie's resignation was not accepted.

The government agency has been accused of using "Wonga-style" tactics in sending out letters from Smith Lawson and Company Recovery Services – a firm it set up – demanding repayment for graduates that were in arrears.

In a written statement, the universities minister, David Willetts, said the government believed 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.

Brodie took up his post in February and met Cable on Thursday to discuss the situation.

In the statement, Willetts said: "Christian Brodie, the chairman of the SLC, made a clear and unequivocal apology on behalf of the company in a discussion with the secretary of state on Thursday 2 July. I also spoke to Mr Brodie on Thursday to investigate what had happened.

"Mr Brodie tendered his resignation as chairman to the secretary of state but this was not accepted. The secretary of state and I have confidence in his leadership of the Student Loans Company.

"We see Mr Brodie having a very important job to do at the SLC to oversee the significant investment the Department for Business, Innovation and Skills (BIS) is making in the company to enable it to transform its products and operations and overall levels of customer service.

"We are both clear that it would be unfair for Mr Brodie to take the blame for a practice that was nearly 10 years old, particularly as he had only been chairman since he joined in February of this year."

The SLC had been issuing letters to graduates who fell into arrears on their repayments under the name of Smith Lawson and Company since 2005, Willetts said. In total, about 309,000 people received letters, which were sent to individuals who had failed to respond to several contact attempts by the SLC.

The practice, which was seen as a "low cost alternative" to referring these graduates to debt collection agencies, was approved in late 2004 by the SLC's board at the time and ministers under the Labour government, Willetts said.

He insisted that the approach was not known to him or Cable before recent media coverage of the issue.

The letters were stopped on 27 June, following the Financial Conduct Authority's recent decision to sanction payday lender Wonga for "aggressive and misleading practices". Willetts said that the SLC had not broken any rules nor charged graduates for getting the letters.

But he added that "it was clear that a public body should be holding itself to the highest standards in the treatment of its customers".

A four-point plan has been drawn up to address the issues raised, and any new plan to deal with debt collection will be cleared by the SLC board and ministers.

"It is important that the government recover taxpayers' money, but it must do so in a way which is fair," Willetts said.

"It must not use misleading tactics to get people to do the right thing. The SLC has apologised and now stopped issuing letters under the Smith Lawson trading name. I will be working with the company to ensure that lessons are learnt to ensure there is no repeat in future."