For the past three years, Elon Musk has steadily elbowed his way into the booming market for in-flight Wi-Fi, with his Starlink service signing up blue-chip carriers from Air France to Qatar Airways to United Airlines.

Alaska Air Group Inc. said on Wednesday it will install Starlink starting next year and Virgin Atlantic in early July reached a deal to use the SpaceX-owned satellite network. Arch-rival British Airways might be not far behind, according to people familiar with the matter, in a deal that would let Musk lock up another flagship airline on the lucrative trans-Atlantic route.