The deal also reportedly includes Costa Rica goalkeeper Keylor Navas heading to Manchester United as de Gea's replacement at Old Trafford.
Should the move become official, De Gea, a 24-year-old Madrid native, would join the club in his hometown as the successor to the departed Iker Casillas.
De Gea had one year remaining with Manchester United, but has not been included in manager Louis van Gaal's squad since the July 29 game against Paris Saint-Germain. Sergio Romero has filled his spot as the team's starting goalkeeper.
Since 2011, de Gea appeared in 131 games for Manchester United. He began his career in Atletico Madrid's academy system before joining the senior squad from 2009 to 2011 where he helped the team win the UEFA Europa League and the UEFA Super Cup in 2010.