This is known as an exploding offer.
So what should you do if you receive an exploding offer?
Exploding offers are anathema to your ability to effectively navigate the labor market. Thus, there is only one thing to do. Treat the offer as a non-offer unless the expiration window is widened.
In no uncertain terms, convey that if the offer is exploding, it’s useless to you.
Example conversation:
I have one big concern. You mentioned that this offer explodes in 48 hours. I’m afraid this doesn’t work at all for me. There’s no way that I can make a decision on this offer within a 48 hour window. I’m currently wrapping up my interview process at a few other companies, which is likely to take me another week or so. So I’m going to need more time to make an informed decision.
If they push back and say this is the best they can do, then politely reply:
That’s really unfortunate. I like [YOUR COMPANY] and was really excited about the team, but like I said, there’s no way I can consider this offer. 48 hours just too unreasonable of a window. The next company I join will be a big life decision for me, and I take my commitments very seriously. I also need to consult with my [EXTERNAL_DECISION_MAKER]. There’s no way that I can make a decision I’m comfortable with in this short an amount of time.
Pretty much any company will relent at this point. If they persist, don’t be afraid to walk away over it. (They probably won’t let that happen, and will come grab you as you’re walking out the door. But if they don’t, then honestly, screw ‘em.)
I was given several exploding offers during my job search. And every time, I did essentially this. Every single offer immediately widened to become more reasonable, sometimes by several weeks.
I want to emphasize, lest I be misunderstood here—what I’m saying is not to just silently let an exploding offer expire, and assume that everything will be fine and they’ll still hire you. They won’t. For exploding offers to be a credible weapon, a company has to have a reputation of enforcing them. I’m saying explicitly call this out as an issue when they make the offer.
Don’t let a company bully you into giving away your negotiating power.
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https://haseebq.com/my-ten-rules-for-negotiating-a-job-offer/
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.