A friend gave me an older iMac. It has a wireless mouse and keyboard. The keyboard works, however, the Bluetooth mouse is stating that it is looking for a wireless mouse. I put new batteries in and have it turned on..

First thing to do is us a USB mouse (any will work) and ensure the computer was properly set per the directions in What to do before selling or giving away your Mac. After you have properly set it up, then follow the steps below if you are still having a problem.

The following will require any USB keyboard and mouse. If you don’t have them, borrow or buy them, they are inexpensive!

1.      Connect the USB keyboard and mouse to the computer.

2.     Open System Preferences – Bluetooth

3.     Un-pair the wireless keyboard and mouse.

4.     Turn off Bluetooth

5.     Restart the computer in Safe Mode per the directions in https://support.apple.com/en-us/HT201262

6.     Restart the computer normally.

7.     Open System Preferences – Bluetooth

8.   Turn on Bluetooth

9.     Re-Pair the wireless keyboard and mouse.

10.  Unplug the USB keyboard and mouse and begin testing the wireless devices again.

You can easily reset the Bluetooth module, this is the method for Big Sur but for older versions you need to search for the appropriate method for the version of MacOS on your computer.

https://www.idownloadblog.com/2021/01/07/reset-bluetooth-mac-tutorial/