Sounds like a problem with the 'phone line to the router. Is the router connected to the master 'phone socket or to an extension socket? If connected to an extension, try connecting the router to the master socket and see if that changes things. If so there is likely to be a problem with the extension circuit.
If above fails, contact BT or your 'phone supplier and ask them to run a check on your 'phone line from the exchange. If this is OK then I suggest you contact your ISP and ask them to investigate the problem.