Kepa, off contract in June, is a priority target for Real in 2018 as they seek to shake up their keeper options. The shot-stopper made his Spain debut this week.
Kepa is keen on the move, says Marca, but is reluctant to risk his World Cup chances.
The goalkeeper's priority is a place in Spain coach Julen Lopetegui's squad for Russia and knows moving to Real would place all his hopes in doubt.
At Real, meanwhile, coach Zinedine Zidane has already approved the move on recommendation of keeper coach Luis Llopis. Llopis previously worked with Kepa at Bilbao.