Visa Inc. has a new offer for small merchants: take thousands of dollars from the card giant to upgrade their payment technology. In return, the businesses must stop accepting cash.

The company unveiled the initiative on Wednesday as part of a broader effort to steer Americans away from using old-fashioned paper money. Visa says it is planning to give $10,000 apiece to up to 50 restaurants and food vendors to pay for their technology and marketing costs, as long as the businesses pledge to start what Visa executive Jack Forestell...