This is a question I see a lot. The short answer is no, the long answer is that even if you would periodically poll the web site it will eventually recycle the app pool anyway.
If need to do background work like this and embed that in ASP.NET you have to create a robust work queue that doesn't break if there are interruptions or crashes because it's going to happen. And that's just good design anyway for long running processes. This might seem like a lot of work but a simple design can take you very far.
The recommended approach is to pull that code into a separate Win32 service because the nature of such workloads don't sit well in REST based architectures.
If all you need is a periodic check, then it might be fine with just having an external script polling the web site but it's a crude way of handling timers.