While Jo Johnson, who was moved from universities minister to transport on 9 January, is not mentioned in the article, he is also thought to have opposed any moves to cut university tuition fees in England from £9,250 a year, having initially sought to link fee uplifts to the results of the teaching excellence framework.
He was Exchequer Secretary to the Treasury from May 2015 before becoming Employment Minister at the Department for Work and Pensions in July 2016.
The Prime Minister's official spokesman said: "Nick Timothy left Downing Street more than seven months ago and he doesn't speak for the Prime Minister or for the Government".
Greening was also the minister who, alongside the then children's minister Edward Timpson, u-turned on the government's bid to put the "exemption clause" in the Children and Social Work Act after a protracted battled with the profession. Greening chose to resign from the Government after refusing another post she was offered.
Damian Hinds, the Member of Parliament for East Hampshire, was educated at St Ambrose College, a Catholic grammar school in Cheshire, before attending Oxford University.
"Greening blocked proposals to reduce [university] tuition fees and refused to hold a proper review of tertiary education", said Mr Timothy.
He wrote: "Young people must be given better choices at 18. He is already touted as a potential future prime minister: if he gets this right, he will be a convincing candidate for the job".
During PMQ's yesterday, Ms Greening reappeared on the Tory backbenches sitting next to anti-Brexit rebel Dominic Grieve.
He denied lobbying for Ms Greening's sacking but described Mr Hinds" promotion as "the bright point in an otherwise limited set of changes'.
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