Visa Inc. said Monday it would buy Plaid Inc. for $5.3 billion, as part of an effort by the card giant to tap into consumers’ growing use of financial-technology apps and noncard payments.

More consumers over the past decade have been using financial-services apps to manage their savings and spending, and Plaid sits in the middle of those relationships, providing software that gives the apps access to financial accounts. Venmo, PayPal Holdings Inc.’s money-transfer service, is one of privately held Plaid’s biggest customers.

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