This is known as an exploding offer.
Follow the directions in the linked article to get rid of them.
https://haseebq.com/how-not-to-bomb-your-offer-negotiation/ (I'll come back here later to extract the relevant quotes, I just don't have much time right now)
But this is even far worse than an exploding offer.
Never accept a verbal offer by itself. A verbal offer is worth nothing. The same goes for a written offer. What he needs is an actual written contract with things like an actual start date, salary details, details of benefits, etc.
If he accepts their verbal offer without anything in return, he'll become their plan B. They'll just interview other candidates, knowing that they have a backup candidate that they can always go back to. Or they will low ball him on the salary and the benefits, knowing that he stopped looking for a job elsewhere.
And one last thing, you said a "headhunter"? Do you mean to say that the person is a 3rd party recruiter? If he's negotiating with a 3rd party recruiter, he's negotiating with the wrong party. He should contact the company directly. The client company may not even be aware of the shenanigans the headhunter is trying to pull on its candidates.