The Trump administration, days before imposing sanctions aimed at the heart of Iran’s economy, is wrestling with a decision critical to its pressure campaign: How hard to push European allies to cut off the country from the global banking system.

Treasury Secretary Steven Mnuchin signaled that the U.S. may not force Belgium-based financial-messaging service Swift to disconnect Iranian banks from the global banking network. The secretary has told foreign governments the U.S. could take a less-confrontational approach, according...