Offer letter sent - but then the old employee has denied to leave the organization

We have offered a job to a seeker but our old employee has denied to leave the organization. What should we do in such a case?

The candidate can be told to go back to his old company and ask for buyout of notice period which new company can pay.
Beyond that new employer cannot do much if old company is not relieving on time.

As far as i comprehend, you mean to say that you had hired a replacement against one of your employees who had resigned from his services. and now your employee has expressed to continue with his services and not exit.
Under such a circumstance, the employer needs to first decide if he wants to retain the services of the exiting employee. If yes, then you can ask the replacement candidate to check with his employer if he can be retained, if not then you can revoke the offer and pay compensation in lieu of it; or make an offer for any other available suitable role for the new candidate.
If the employer wants to let the old employee go, then just accept his resignation and proceed for his exit; and make way for the new candidate.
Regards
Rahul Chhabra

There are couple of things to be considered before taking a decision. You had offered a job-seeker and basing on it he/she might have resigned his/her existing job to join your company. What would be their position in case you go back on your word ? Secondly, what if the concerned employee again puts in his/her resignation? What is the guarantee that he/she may not do it again? How long will the said employee continue, what's the guarantee? Since the resignation has been accepted you need to explain to the employee that you have acted on the same and taken a replacement and move forward. That's my view.
Thanks and Regards

Please give more details about the scenario. Has your company accepted the employees resignation and notified the person the date of releasing him/her? Is there a service rule whereby a person that has resigned can withdraw the resignation?