Iran's Islamic Revolutionary Guard Corps (IRGC) said it had targeted an air base used by US forces for an attack on southern Iran, without specifying the location.
Meanwhile Kuwait, which hosts a US base, said its air defence system was "confronting hostile missile and drone attacks" without providing further detail.
Trump urged his critics to "sit back and relax" in a post on Truth Social early on Monday, saying it would "all work out well in the end". He said Iran "really wants to make a deal, and it will be a good one for the USA".
The strikes mark the latest exchange between the two sides after negotiations on a deal to end their months-long war failed to advance over the weekend, with US media reporting President Trump had requested changes to its terms.
