The winger has been linked with a Premier League return, and his former club appears to be the frontrunner
The winger has fired 105 goals in 251 matches at Real Madrid, having also won two La Liga titles to go with those four European triumphs.
However, the 31-year-old has fallen out of Zinedine Zidane's plans as he made only 16 appearances last season in all competitions.
Bale has been heavily linked with a departure this summer, having also been frequently rumoured to be on the move several times in recent seasons.
Manchester United and Tottenham have been said to be interested and, on Tuesday, Bale's agent confirmed that a return to Spurs is his preferred destination.
"Gareth still loves Spurs," Barnett told BBC Sport Wales. "We are talking [Spurs, Real and Bale's camp]. It's where he wants to be."
Bale originally broke through with Spurs in 2007-08 having made the move from Southampton the summer prior.
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He rapidly became a key figure at White Hart Lane before truly emerging as a world-class star in 2012-13 as he scored 26 goals, including 21 in the Premier League.
Bale won both the PFA Players' Player of the Year and Young Player of the Year that season, also adding the FWA Footballer of the Year Award to join future team-mate Cristiano Ronaldo to become only the second player to win all three in the same season.
Internationally, Bale has been a mainstay for Wales for nearly a decade-and-a-half, scoring 33 goals in 85 games for his country.
He featured in recent Nations League clashes with Finland and Bulgaria, with Wales earning 1-0 wins in each match.
Tottenham started the Premier League season with a difficult defeat, falling to a 1-0 loss to Everton to begin their first full season under manager Jose Mourinho.
Real Madrid, meanwhile, are fresh off a La Liga title and will begin the defence of that crown on Sunday with a visit to Real Sociedad.