You will have to litigate to make the company prove the damage can be pinned on you. To know what the company is able to prove, you will need to have access to all the prior rental records. Somewhere at some time the damage was caused. The trail starts with the documents in their possession.
The other poster is correct. To prove their case, the rental company will have to file suit. You need to look at the rental agreement to see if there are any limitations on where and when such an action can be filed. Also, you should talk to your insurance agent about this possible claim so they can put your appropriate insurance carrier on notice. Good luck.
DISCLAIMER: Please be advised that this post is not intended to constitute legal advice and is for informational purposes only. This posting in no way creates an attorney client relationship. You should contact an attorney to protect your interests.
You have the right to decline their offer to pay them. They have the right to sue you. When and how will be set forth in your rental contract with them. Whether they prevail will depend upon the facts presented. From the facts you have presented so far, your defense appears presentable.
No information provided in response to these questions can be relied upon in any way without further personally consulting with Attorney Kerrigan and Attorney Kerrigan consulting personally with you regarding your specific legal situation.