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The draw left champions Barca top on 36 points after 17 games with Real also on 36, five clear of third-placed Sevilla.
Lionel Messi celebrated his sixth Ballon D'Or award with a brilliant hat-trick as Barcelona moved back top of La Liga with an emphatic 5-2 victory over Real Mallorca on Saturday.
The draw leaves Barca still leading with 35 points from 16 games but Real Madrid, on 34, can take their place at the top if they win at Valencia on Sunday.
Real Madrid climbed to the top of the standings with 14 points from six games, one point and one place ahead of Atletico Madrid, whom they face on Saturday.
Bale has a long and successful record playing at Villarreal, scoring there on his Real debut in 2013 after completing his 100 million-euro move from Tottenham Hotspur and netting in a 3-2 win in 2017 and a 2-2 draw in 2018.
Real Madrid are top with 15 points from seven matches and Atletico stayed third with one point less.
Strikes by Luis Suarez and Junior Firpo either side of halftime gave the Spanish champions their first win of the season.
Karim Benzema scored twice for Real Madrid in a 3-2 win against Levante with Eden Hazard making his La Liga debut while Barcelona breezed to a 5-2 win against Valencia with more magic from 16-year-old Ansu Fati.
The Catalans started without all-time top scorer Lionel Messi as well as teenage sensation Ansu Fati and fell behind in the second minute.
The victory takes Barcelona up to fourth with 10 points after six games, one point behind surprise leaders Granada, Athletic Bilbao, and Real Madrid.
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