The two top makers of the highly specialized swabs used to test patients for the novel coronavirus are straining to keep up with the demand, even as both the Italian and U.S. governments are working with them to increase production, including at a key manufacturing site in the midst of Italy’s outbreak.

The nasopharyngeal swabs required for the coronavirus tests are quite different from your standard Q-tips — and the exploding need for them has created a bottleneck in the soaring demand for d...