The email, seen by Times Higher Education, was sent to staff at the University of Birmingham's School of Philosophy, Theology and Religion by Helen Beebee.
She writes that she "CANNOT STRESS ENOUGH" how important it is that all academics "try as hard as we possibly can" to recruit postgraduates for the coming year.
Professor Beebee warns in the memo, sent at the end of July, that Birmingham's College of Arts and Law, of which her school is part, is facing a financial penalty of "getting on for ?1 million" from the university for under-recruitment.
"That is a lot of money per school - and the brunt of this penalty may well be borne by those schools that fall the furthest short," she writes.
As a result, she urges academics to be "VERY generous in your judgement about whether the candidate is capable of undertaking the programme applied for", adding that "we simply cannot afford to be very choosy".
Malcolm McCrae, chair of the UK Council for Graduate Education, described the email as "unfortunate and ill-considered".
"It is well known that students whose capabilities are not on a par with the demands of the programme they are following always turn out to be much more work, accentuating the pressure to compromise academic standards in an effort to get already recruited students through to...completion," he said.
He added that when such students started jobs, it would "quickly become apparent that they aren't up to the quality their qualification says they should be".
This inevitably would have a negative effect on the academic reputation of both the institution and UK higher education as a whole.
But he said that the current "mood music" of increased competition and fears that postgraduate recruitment could be hit by the rise in undergraduate fees meant that universities were desperate to maintain their "market share".
Professor Beebee also writes in the email that "NOBODY" should reject a PhD candidate on the grounds that they were too busy to supervise them.
"If anyone is carrying too high a burden because of increased (postgraduate) recruitment, we will look at ways of reallocating work once the academic year starts," she says.
In a statement, Birmingham does not address the content of the email directly, but says: "The quality of our postgraduate students is reflected in our standing as a leading global university."
It adds that recruitment targets with financial rewards and penalties have been set at the college level at the institution since 2008.