The Student Loans Company is facing calls to compensate more than 300,000 graduates sent misleading "Wonga-style" debt collection letters, as it emerged that the business secretary, Vince Cable, is to meet the agency to question its tactics.

The letters, sent between 2005 and June of this year, purported to be from Smith Lawson and Company Recovery Services, acting on behalf of their government "client". In fact, the firm was merely a subsidiary of the Student Loans Company, which was only revealed in tiny print at the bottom of the letters.

Speaking in parliament, Lord Stevenson of Balmacara said that when similar practices were undertaken by the payday lender Wonga the firm agreed to make "redress to those affected by letters of this type with significant amounts of money".

"Will you confirm that all students who have been similarly affected in this way will also receive redress?" the Labour peer asked the government.

The Student Loans Company stopped sending out the letters on 27 June just days after it emerged that Wonga used fake law firms to threaten people who owed the payday lender money. Wonga has agreed to pay £2.6m in compensation to customers and apologised "unreservedly" for its failings.

Lord Ahmad, a government business spokesman in the Lords, said Cable would now look at "what other remedial action is necessary" in relation to the Student Loans Company.

But the peer insisted there was a "difference" between the tactic adopted by the government agency and by Wonga.

"First of all there were no charges made for these particular letters that were issued and secondly the actual name of Smith Lawson was set up as a trading name and was in accordance with procedures registered with the intellectual property office," he said.

"But the secretary of state and the minister responsible are looking at this situation and any further action that needs to be taken I will let the House know."

Liberal Democrat peer Lord Paddick told peers: "Companies sending threatening letters falsely representing themselves as debt collection agencies is something we might expect of payday loan companies but not something from a government agency."

In a statement, the Student Loans Company said it has merely "suspended" use of the Smith Lawson letters and it may revive the debt collection "branding" at some point.

"The Smith Lawson collection process has been used since 2005 to support collection activities and was mainly used after the initial in-house collections process had been exhausted for customers who had persistently defaulted on their loan repayments and were in arrears for a minimum of three months," it said.

"The Student Loans Company has suspended all use of Smith Lawson branded correspondence. The company had reviewed its Smith Lawson letters following discussions with the Office for Fair Trading in March 2014 and had agreed to implement their suggested changes.

"The issuing of any revised Smith Lawson correspondence has still to be agreed."