LONDON—The U.K. government is in talks with the U.S. and the European Union over a possible exemption to Iranian sanctions that would allow BP PLC to restart a North Sea natural gas field partially owned by a Tehran-controlled company.

The move comes as the U.S. and its allies have welcomed the elevation of a new Iranian president earlier this year as a possible opportunity to re-engage Iran in talks about its nuclear program. But the effort also comes amid increasing worry in the U.K. over Britain's natural gas supplies. Amid...