NBC, Jimmy Fallon and Seth Meyers intend to pay staffers of the network’s “Tonight” and “Late Night” shows three weeks’ of wages while the programs are sidelined due to the writers strike, according to two people familiar with the matter.

NBC plans to pay two weeks of salary to staffers while each late-night host will pay a third week out of their own pockets, according to these people. Healthcare for the shows’ employees will be paid through September. Staffers were informed Wednesday morning during production calls, these people say, with Fallon and Meyers taking part personally to discuss the matter with his staff. The hosts typically do not participate in those early-day meetings.

NBC declined to make executives available for comment.

The moves suggest the network and the hosts would like to return to the air sooner rather than later. In the writers strike that took place in 2007 and...