Fourth-placed Celta stayed two points behind Real, who are third, thanks to a 2-0 win at Granada on Sunday while Villarreal are a point adrift of the Galicians in fifth after they won by the same scoreline at Real Sociedad.
By contrast Europa League champions Sevilla and Valencia, backed by the huge wealth of Singapore owner Peter Lim, are trailing eighth and ninth respectively after they drew on Saturday and in danger of missing out on a berth in Europe.
Deportivo La Coruna, 2-0 winners at home to Eibar on Saturday, occupy the sixth and final European qualification place ahead of Athletic Bilbao in seventh after the Basques beat bottom club Levante 2-0 on Sunday.
Coached by Argentine Eduardo Berizzo, Celta handed Barca one of their biggest defeats in recent years when they hammered the Spanish and European champions 4-1 in September.
Villarreal stunned visiting Real last weekend when they beat them 1-0, having seen off Atletico by the same scoreline at the Mestalla in September.
Both teams play entertaining, attacking football and are managing to compete despite having significantly smaller budgets.
Celta's squad has a market value of 75 million euros (55 million pounds), according to website Transfermarkt.com, while Villarreal's is worth 136 million.
Both are dwarfed by Valencia's 275 million, with Sevilla's worth 191 million.
Villarreal finished sixth last term with 60 points to qualify for the Europa League and they are through to the last 32 of the continent's second-tier club competition this season.
They already have 30 points and, like Celta, are well placed to secure another lucrative berth in Europe.
"We are very pleased and it leaves us in a very good situation to enjoy Christmas," Villarreal coach Marcelino told a news conference.
Celta just missed out on a European spot last term when they finished eighth but their current position would earn them a place in Champions League qualifying and their highest finish since 2002-03.
"Our success is based on the effort the players put in and the fact the team is totally a team," Berizzo said.
(Writing by Iain Rogers, editing by Tony Jimenez)
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