The 777-100 was among the proposal of Boeing back in 1978. It was actually a tri-jet 767.
More recently, the 777-100X is a project which has been core part of the whole 777X development, with eventually 777-200LR and -300LR going ahead, but still there was a true request from American Airlines and Singapore Airlines.
Even more recently, this is a proposal to Singapore Airlines as a replacement for its A310s. It is still unsure whether the aircraft should be a simple shrunk 777-200 or an ultra-long-haul aircraft.