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The country remains on thin ice with the governing organisation, currently sits 199th in the FIFA rankings and has still never qualified for a football World Cup.
A 1-0 victory for Diego Simeone's side cuts the gap from five points to two behind Barca, who will restore their advantage with a win over Eibar later on Sunday.
The German champions -- who sit in an unfamiliar third place in the Bundesliga -- would be ready to pay a release clause of 80 million euros ($90 million) to bring the France international to Bavaria.
The win was Atletico's first away from home in La Liga in nearly three months.
Luka Modric sees off French World Cup stars to end Lionel Messi, Cristiano Ronaldo grip on Ballon d'Or
Cristiano Ronaldo was second, with France and Atletico Madrid striker Antoine Griezmann completing the podium.
Monaco, which had arrived with no chances of advancing, missed a late penalty as former Atletico forward Radamel Falcao shot wide.
Barca produced another fightback as Dembele, on as a substitute, slid home a cool finish in the 90th minute to earn the visitors a 1-1 draw.
Following the win, Alaves temporarily holds the third spot in La Liga with 23 points, on goal differential behind Atletico Madrid, while Huesca remains dead last in 20th with six points.
Godin pulled a muscle in the 64th minute on Saturday as Inaki Williams gave Bilbao a 2-1 lead at the Wanda Metropolitano, but Atletico's captain stayed on.
Dortmund handed Atletico their heaviest defeat under Diego Simeone with a 4-0 victory 13 days ago but this contest was almost a mirror image, the hosts in control from start to finish.