I have recently got a new SSN. But I already own a student credit card from the past one year, issued by a financial institution without a credit check. As I didn't have an SSN when I applied for this card, my SSN is not currently associated my the credit card. If I now link my new SSN to the credit card, what all information will the financial institution report? I have made a few late payments a few months back. Will that reflect in the credit report/history/score?
That's really going to be up to the issuer of the card as to how they handle this. The credit agencies are just reporting what's been given to them. If you have late payments with that card then you might be better off not having it reported.
I'm not sure how any credit card issuer would issue a card without a social security number, even if they don't do a "hard pull" (actually pulling a score or credit report) on your credit, because it's one of the only ways they have to verify who you are. It isn't uncommon for some issuers to issue low-limit cards without doing a hard pull, but issuing with a verifiable SSN? Odd.
Be that as it may, everything's going to depend on whether the card issuer decides to retroactively report your history if/when there is a link between the account and your SSN, and that's a call they make. I am unaware of any particular rule that the credit bureaus have on this, but since the bureaus themselves strive to be complete in their reporting (since having the most complete info is what justifies the fees they charge), they may require creditors to provide a borrower's full history.
In the end, you will just have to wait and see. If the card issuer didn't care to collect an SSN from you when you opened the account then there's no particular reason you have to give it to them now, and you don't want to run the risk of those late payments showing up and dinging your credit score, so one option is to not give them your SSN and let sleeping dogs lie.
I hope this helps.
Credit reporting agencies know they are working with data that is not entirely clean and complete. They do not require an exact SSN match to combine records into a credit report. Nor do all accounts with the same SSN end up in the same credit report.
It is possible that even if nothing changes, that the account without SSN may show up on your credit report based on name and address matching. The only way to know for sure would be to request your credit reports.
Years ago Amex misread or mis-keyed my SSN, so that the record they gave to the credit reporting agencies did not have my SSN. That account was on my credit report.
Years ago I read an article where a person tried to open an account and the account was denied because an account with the same SSN existed at the financial institution. Person complained to the credit monitoring agency why they did not catch that someone else had taken out credit with their SSN. Monitoring company responded that account was not on his credit file. SSNs are not an absolute key to credit files.