Wells Fargo plans to move to data centers owned by Microsoft and Google over several years; Morgan Stanley is also working with Microsoft. Bank of America has saved $2 billion a year in part by building its own cloud. Goldman said in November that it would team up with Amazon Web Services to give clients access to mountains of financial data and analytical tools.

Cloud services enable banks to rent data storage and processing power from providers including Amazon, Google or Microsoft, which have their own data centers dotted around the globe. After moving to the cloud, banks can access their data on the internet and use the tech companies’ computing capacity when needed, instead of running their own servers year-round.

Seeing a big opportunity to sell cloud-computing services to Wall Street, some tech giants have hired former bankers who can use their knowledge of the rules and constraints under which banks operate to pitch the industry.

Scott Mullins, AWS’s head of business development for financial services, previously worked at JPMorgan and Nasdaq. Yolande Piazza, vice president for financial services at Google Cloud, is the former chief executive of Citi FinTech, an innovation unit at Citigroup. Bill Borden at Microsoft and Howard Boville at IBM are Bank of America alumni.

Cloud providers are “moving at a much faster development pace when you think of security, compliance and control structures,” compared with individual banks, said Mr. Borden, a corporate vice president for worldwide financial services at Microsoft. The cloud, Mr. Borden and the other executives said, enables companies to increase their computer processing capabilities when they need it, which is much cheaper than running servers on their own premises.

But glitches do occur. One week after Goldman teamed up with Amazon, an AWS outage halted webcasts from a conference hosted by the bank that convened chief executives from the biggest U.S. financial firms. The glitch also caused problems for Amazon’s Alexa voice assistant, Disney’s streaming service and Ticketmaster. AWS and its competitor, Microsoft Azure, both had outages recently.

Banking regulators in the United States, including the Federal…