Question: I asked to close my card after clearing dues. The issuer says closure can take a few weeks. Is there a deadline?
Answer: Yes. The issuer must honour a credit-card closure request within seven working days after dues are paid and must notify the cardholder. Delay beyond that attracts a penalty of INR 500 per calendar day, provided there is no outstanding amount.

Supporting evidence:
title: rbi.org.in/BS_ViewMasDirections.aspx
quote: Any request for closure of a credit card shall be honoured within seven working days by the credit card-issuer, subject to payment of all dues by the cardholder Subsequent to the closure of credit card, the cardholder shall be immediately notified about the closure through email, SMS, etc Failure on the part of the card-issuers to complete the process of closure within seven working days shall result in a penalty of ₹500 per calendar day of delay payable to the cardholder, till the closure of the account provided there is no outstanding in the account
context: entified by the principal cardholder under both personal and business credit card categories. Add-on cards shall be issued with the clear understanding that the liability will be that of the principal cardholder. Similarly, while issuing corporate credit cards, the responsibilities and liabilities of the corporate and its employees shall be clearly specified. The liability of the corporate/business entity shall form part of its assessed credits.

5. Closure of Credit Card (a) iii Any request for closure of a credit card shall be honoured within seven working days by the credit card-issuer, subject to payment of all dues by the cardholder. Subsequent to the closure of credit card, the cardholder shall be immediately notified about the closure through email, SMS, etc. Cardholders shall be provided option to submit request for closure of credit card account through multiple channels such as helpline, dedicated e-mail-id, Interactive Voice Response (IVR), prominently visible link on the website, internet banking, mobile-app or any other mode. The card-issuer shall not insist on sending a closure request through post or any other means which may result in the delay of receipt of the request. Failure on the part of the card-issuers to complete the process of closure within seven working days shall result in a penalty of ₹500 per calendar day of delay payable to the cardholder, till the closure of the account provided there is no outstanding in the account. (b) If a credit card has not been used for a period of more than one year, the process to close the card shall be initiated after intimating the cardholder. If no reply is received from the cardholder within a period of 30 days, the card account shall be closed by the card-issuer, subject to payment of all dues by the cardholder. The information regarding the closure of card account shall also accordingly be updated with the Credit Information Company/ies within a period of 30 days. (c) Subsequent to closure of credit card account, any credit balance available in credit card accounts shall be transferred to the cardholder’s bank account. Card-issuers shall obtain the details of the cardholder’s bank account, if the same is not available with them.

6. Interest rates and other charges (a) Card-issuers are directed to be guided by the instructions on interest rate on advances issued by the Reserve Bank as amended from time to time, while determining the interest rate on credit card dues. Interest charged on cred