The Daily Telegraph claims that negotiations between the two parties are officially underway, with the Wales forward in line for an increase on his £300,000 ($400,000) per week wages. A new deal could be reached within three weeks, the paper says.
Bale has three years left on his contract at Real, where he has won two Champions League titles and one Copa del Rey.
The Wales international said last month that he was "very happy" in the Spanish capital after scoring 19 goals in 23 league appearances for Real last season.
"I feel more integrated, I have more personal relationships with my teammates and that always gives you more happiness and that reflects on my game," he was quoted as saying by Sky Sports.
"I'm enjoying myself a lot in Madrid and I hope this continues for many years.I want to be better each season and win many titles along the way."
With three goals for Wales in four appearances, Bale is also the joint-top scorer at Euro 2016. The 26-year-old is preparing for his country's quarter-final showdown with Belgium in Lille on 1 July.
He scored the only goal of the game the last time the two teams met in a qualifier in June last year in Cardiff.
"From that night, we have gone up to another level," Bale was reported as saying by BBC Sport on 29 June. "We're like their bogey team, aren't we? Hopefully that edge gets into their heads, you never know.
"We will definitely draw on those qualifiers, it gives us more knowledge on them. But we understand what they are about, how they work, and we will hone our game plan to try and stop them but help our attack."