The 26-year-old scored in each of his country’s three matches as they topped Group B, finishing above England, after helping Madrid win an 11th Champions League title last season.
Bale has attracted the interest of Manchester United in recent seasons but, according to the Daily Mail, Madrid owner Florentino Pérez is prepared to offer him seven more years at the Bernabéu on one of the most lucrative contracts in world football.
While Cristiano Ronaldo remains a prized asset in the Spanish capital, Madrid executives are planning for life without the Portuguese with Bale likely to become the “face” of the club in the coming seasons.
The former Southampton and Tottenham Hotspur forward joined the club for a world-record £85m in 2013 and has scored 58 goals in 123 appearances for Los Blancos since leaving England.
Despite speculation to the contrary, Bale is understood to be happy in Spain and is expected to welcome the negotiations when he returns from international duty next month.
Meanwhile, Madrid have activated a buy-back clause to take Spain striker Alvaro Morata back from Juventus, the European champions announced on Tuesday.
"Real Madrid have informed Juventus of their decision to buy back the registration rights on the player Alvaro Morata, who will join up with the first team for pre-season training under the command of Zinedine Zidane," said a Real statement.
Join our new commenting forum
Join thought-provoking conversations, follow other Independent readers and see their replies