You can set contract notifications by going on Setup -> Settings -> Cron Job -> Contracts.
Only contract creator or admin will receive expiration notification for a specific contract.
If contract is expired the list table row will be filled with red background.
You have option to receive contracts expiry notifications additionaly to email. Check the email template from Setup-> Email templates.
If you dont want to receive emails for contract expiration notification you only need to disable this email template.