Shaw was knocked out when his head thudded to the floor sickeningly after an aerial clash with Carvajal in England's 2-1 Uefa Nations League win.
The Sun says Spain defender Carvajal, 26, tried in vain to find Shaw after Saturday's tie.
The Real Madrid star went into the England dressing room but Shaw was being treated elsewhere.
So Carvajal asked Spain keeper De Gea for the phone number of his Manchester United team-mate.
Real centre-back Ramos simply said: "Get well soon."