Hunt told media immigration was required to boost growth, adding there had to be "a long-term plan if we're going to bring down migration in a way that doesn't harm the economy".
Besides, the British government's department of education could also raise objections, as the contemplated cut-down would increase state funding of universities, who otherwise benefit from high fee-paying international students.
The chairman of Whitehall's Migration Advisory Committee, Brian Bell, said in an interview to BBC that the idea said to be in Sunak's in-tray could "send many universities over the edge".
"Most universities for most courses lose money on teaching British students and offset that loss by charging more for international students," Bell emphasised.
Indians constitute the largest contingent of foreign students in the UK at present. If the restriction being mulled over by Sunak becomes policy, they are likely to be the worst sufferers.
New figures disclosed net migration had exceeded half a million - an increase of 300,000 in a single year. Among additional, the steps Sunak is believed to be considering is a clamp down on visas for dependents of overseas students.
Sunak's hard line Home Secretary, Suella Braverman, previously expressed concern about foreign students "bringing in family members who can piggyback onto their student visa". According to her, they undertake "substandard courses in inadequate institutions".
Earlier, in an interview to The Spectator magazine, she accused Indians of being the biggest illegal over-stayers in the UK.
Braverman is in fact opposed to freer movement of people from India to the UK, which is one of the Indian government's demands in the current negotiations over a free trade agreement with the UK.
However, the right-wing, inward looking Conservative party are happy about their 42-year-old new Prime Minister contemplating curbing the intake of foreign students to reduce immigration. At the same time, some of his MPs are already beginning to give up on him.
One such lawmaker who won from a constituency that was formerly an opposition Labour party stronghold in the north of England, commented to the pro-Conservative Daily Mail newspaper: "Rishi would be a decent project manager. Or senior civil servant. But he's not a natural Prime Minister. He's just not the sort of guy you want to follow out of the trenches."
Meanwhile, the UK's National Grid (LON: ) issued its first emergency warning about a winter blackout. And inflation is soaring. "But", the Daily Mail, highly influential among Conservatives, said: "The Prime Minister was absent. Or, if not exactly absent, busy elsewhere."
Sunak recently visited Kiev to pledge 125 anti-aircraft guns to Ukraine in its fight against Russia.
As economic hardship bites, a majority Britons are seemingly beginning to tire of the war, which wasn't the case in the summer.
In the Mail's opinion: "After the unrelenting bombast of the Boris Johnson years, and the frenetic mayhem that was the month of Truss, such reflective disengagement may be welcome... But the country is experiencing a growing sense of crisis. And what it needs isn't a manager, but a leader."
It headlined elsewhere: "Dire warning for 'tainted' Tories (Conservatives) as bombshell polling reveals Sir Keir Starmer (Labour leader) is ahead of Rishi Sunak on 11 out of 12 key issues - including cost of living, NHS (National Health Service), immigration and Brexit."