The Spanish giants are set to spend big in the summer with Real chief Florentino Perez ready to back Zinedine Zidane in the transfer market.
According to Spanish publication Diario Gol, Bale has put forward his transfer suggestions to Perez and Alli tops his wishlist.
The Tottenham midfielder has been linked with a move to the Spanish capital in the past but committed his long-term future to the club by signing a new six-year deal last summer.
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However, Bale, who moved from Tottenham in 2013, believes the 21-year-old would be a huge hit at the Bernabeu.
Alli has been in sensational form for Mauricio Pochetino’s side this season, scoring 20 goals and registering six assists in all competitions.
The midfielder’s impressive displays on the international stage for England have also seen him attract interest from the likes of Paris Saint-Germain and Barcelona.
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But Real are thought to be his biggest admirers with Bale pushing for the club to make an ambitious move for him in the transfer window.
Midfielder James Rodriguez has been widely tipped to leave Real in the summer and Alli would make an ideal replacement for the Colombian.