We study American swaptions in the linear-rational (LR) term structure model introduced in Filipović et al. [J. Finance., 2017, 72, 655–704]. The American swaption pricing problem boils down to an optimal stopping problem that is analytically tractable. It reduces to a free-boundary problem that we tackle by the local time-space calculus of Peskir [J. Theoret. Probab., 2005a, 18, 499–535]. We characterize the optimal stopping boundary as the unique solution to a non-linear integral equation that can be readily solved numerically. We obtain the arbitrage-free price of the American swaption and the optimal exercise strategies in terms of swap rates for both fixed-rate payer and receiver swaps. Finally, we show that Bermudan swaptions can be efficiently priced as well.