I think you could call the second company, tell them you have already got an offer, and ask them if they could quicken their process.
You should obviously accept the first offer. But that doesn't mean you must go to work at once. You can have a reason (for example, you are ill recently) to delay the start date to one or two weeks later. Good Luck!
Hi, everybody, just get offer letter from one company, but another company also said will give me offer in next few days so what is the best way for me?