Some MPs fear that Offa could compromise universities' independence in running their own admissions.
Offa said Exeter has historically performed below targets for students from state schools and from low income backgrounds.
This is 10% greater than the level of support expected at English universities by the Office for Fair Access (OFFA).
The new director of Offa will replace the current head, Sir Martin Harris, who is stepping down later this year.
Professor Les Ebdon, the new head of the Office for Fair Access (Offa), has urged universities to set themselves "challenging" admissions targets.
Prof Ebdon has re-stated his continued interest in running Offa and says he is awaiting the decision of the secretary of state.
Ysgol Clywedog was opened after three schools - Bryn Offa, St David's and the Groves were merged to create two new schools.
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The university was named with Plymouth College of Art among colleges which failed to meet targets in figures from the Office for Fair Access (Offa).
MPs on the Business, Innovation and Skills Committee opposed Prof Ebdon's appointment to the Office for Fair Access (Offa), but Mr Cable rejected their attempts to block him.
Research from the access regulator Offa indicates the richest 20% of young people are seven times more likely to enter the most selective institutions than the poorest 40%.
He accused Prof Les Ebdon, director of the Office for Fair Access (Offa) and Alan Milburn, the government's social mobility tsar, of "standing firmly in favour of manipulation".
Offa's access agreements will also put an emphasis on universities evaluating and measuring the success or otherwise of their outreach work, so that they can show what works.
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Prof Les Ebdon, director of the Office for Fair Access to Higher Education (Offa), said the body's new guidance to universities would emphasise the importance of "sustained, targeted outreach".
Offa has always had the power to impose fines on universities that fail to boost numbers of students from poorer backgrounds, but has never used it in its history.
The report from MPs expressed concerns about the preferred candidate's suitability, saying that they "struggled to get a clear picture of Professor Ebdon's strategy for the future of Offa".
MPs on the Business, Innovation and Skills select committee voted against accepting the nomination of Prof Ebdon as the new head of Offa - and called for the recruitment process to begin again.
These agreements are be reviewed and approved by the Office For Fair Access (Offa) each year, with institutions that fail to meet their agreed targets on recruitment and retention facing fines or losing the right to charge higher fees.
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