I have a private key of a bitcoin wallet . Is there is any way i can check balance and make transaction using bitcoin core without importing private key.
You can check and watch the balance by adding the address corresponding to the private key as a watch-only address, but it's impossible to make a transaction in Bitcoin Core without importing the private key.
You can use an airgapped machine to derive the address and import it as a watch-only address to another instance. You could also use an airgapped machine or hardware wallet to sign a raw transaction offline.