The highly-rated 23-year-old has become a regular with the La Liga outfit having worked his way into the first team following a season-long loan at Real Valladolid in the 2015-16 campaign.
Kepa's form has also earned him a first cap with the Spain national team in Saturday's 5-0 thrashing of Costa Rica, while it is believed that Los Blancos are now looking at making a bid for his signature as they eye a long-term replacement for Keylor Navas.
The youngster's contract at the San Mames is set to expire in the summer and he has so far appeared reluctant to tie himself down.
Despite the speculation, Kepa has spoken out to insist that he has no desire to leave Bilbao at this moment in time.
He said: "I'm happy at Athletic, I'm calm and I'm only focused on working hard for my club.
"I know that people talk about me because I have family and friends who read the papers and listen to the radio. Obviously I'm not blind to it, but I'm calm."
Kepa added: "Future? The rumours don't make me nervous. I am what I am because of what I do, not because of what they say."