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Before applying for credit cards, I was typically opening a checking account to have a good banking relationship. This method almost worked but after some years, I ended up having several checking accounts that require e.g. monthly ACH or minimum monthly balance to avoid charging monthly fee.
I may want to get rid of some checking accounts but does it impact my banking relationship? For example, if in future, I would plan to apply for a loan, is it sufficient to have a credit card from one specific bank to consider it as a good banking relationship?
That depends on the bank and which product(s) you are targeting. Smaller FIs that still perform their UW manually may take a look at your depository accounts and count them as part of a favorable decision per their criteria. Larger FIs that are automated mostly look at depository accounts and lending products as two separate categories. If it were me, I would close all of the extra account you don't need or use, unless you alread have a specific reason in mind to keep that account going. I think in most instances, it doesn't really matter. It's not so much of presence of positive depository accounts with them, it's more so the lack of negative that really matters.
If you opened so many accounts that you're thinking of closing some, it's likely that the accounts you're targeting for closure had token deposits and little to no activity on them. Those types of accounts probably had a mediocre effect at best on getting approved for a card and will likely have no effect on credit approval if you close them. Some lenders do respond favorably to deposit accountholders but they clearly see the difference in someone who is using the account versus parking a little money. Oftentimes, parking a small amount of money isn't worth the effort since that doesn't constitute a full banking relationship.