The Statute of Limitations for filing of claims varies from state to state. That is the time frame within which a claim must be brought or it will be forever barred. Looking at your avatar, I presume you are in Pennsylvania.
According to 42 Pa. C.S.A. § 5524, a two-year limitation applies in this type of case.
"The following actions and proceedings must be commenced within two
years:
(1) An action for assault, battery, false imprisonment, false arrest,
malicious prosecution or malicious abuse of process.
(2) An action to recover damages for injuries to the person or for the
death of an individual caused by the wrongful act or neglect or
unlawful violence or negligence of another.
(3) An action for taking, detaining or injuring personal property,
including actions for specific recovery thereof.
(4) An action for waste or trespass of real property.
(5) An action upon a statute for a civil penalty or forfeiture.
(6) An action against any officer of any government unit for the
nonpayment of money or the nondelivery of property collected upon on
execution or otherwise in his possession.
(7) Any other action or proceeding to recover damages for injury to
person or property which is founded on negligent, intentional, or
otherwise tortious conduct or any other action or proceeding sounding
in trespass, including deceit or fraud, except an action or proceeding
subject to another limitation specified in this subchapter".
However, you should be aware that his insurance carrier has the right to disclaim coverage if there is late notice to them of the accident. It appears that he wants to pay you privately to avoid his rates going up and this most likely will mean that he has not and will not advise his carrier of the accident. As a result, he is running the risk that they will disclaim coverage if he does not notify them of the accident.
For your sake, I would recommend filing with his carrier yourself if no check is forthcoming by the end of the week. You should also put your own carrier on notice and give yourself the option of being reimbursed by your policy and then having your carrier get your deductible back in a subrogation action. Be mindful of the fact that many subrogated claims are settled for less than 100 cents on the dollar and your deductible reimbursement would be adjusted downward accordingly.
Unless this is a friend of yours who hit you, you owe the other driver nothing in terms of a favor and the preeminent concern should be you getting full reimbursement for the damages. Avoid the risk of the driver not paying and his carrier disclaiming by placing them on notice yourself if not paid in full by this weekend.