Real Sociedad have not reached the Champions League for a decade but Imanol Alguacil's side can qualify for Europe's premier competition with just two more points from their remaining two La Liga games.
The Basque side, fourth, visit Diego Simeone's Atletico Madrid on Sunday, hoping to book their place before a potentially nervy final match of the season.
Villarreal, fifth, are five points behind La Real and must win their last two matches to stand a chance of reaching the Champions League.
Atletico were dealt a blow on Wednesday when they threw away a three-goal lead on Espanyol and now trail rivals Real Madrid, in second, by a point.
Real Sociedad will be without playmaker David Silva, after he sustained a calf injury this week, ruling him out of their final two league games.
The 37-year-old playmaker, who has signed a new contract until 2024, has been a key figure in La Real's fine season and prospective return to the top level of European football for the first time since 2013.
However, another veteran player, Asier Illarramendi will not continue on beyond the end of his deal, expiring in June.
The 33-year-old wrote an emotional letter this week detailing his goodbye and posed for photos with his team-mates and the Copa del Rey the side won in 2021.
That trophy, their first major trophy since lifting the same cup in 1987, was a sign the Basque team are on the rise again.
Alguacil has played a key part in their impressive recent rise.
A player for La Real in the 1990s, the coach took over in 2018 and has moulded a skilful, fun side, but one which always works hard.
Even without Silva, Real Sociedad have the players to get them over the line against Atletico and Europa League finalists Sevilla.
Defensive midfielder Martin Zubimendi is craved by Barcelona as a potential Sergio Busquets replacement, and he excelled in their win at Camp Nou last weekend -- their first since 1991 in La Liga.
That victory, and another over struggling Almeria on Tuesday took them to the brink of accomplishing their objective, which they could do against Atletico.
Winger Takefusa Kubo struck the winner against Almeria and continued his own superb form this season.
"Here, if you relax, the next day you're on the bench," said Kubo on Tuesday.
"The other day, the new players who came in showed they had the level to beat Barca.
"You are unlucky not to be able to see the training sessions, I would like people to see the level of demand on us that there is in Real's training sessions."
The former Real Madrid and Barcelona winger said he would watch Villarreal's game against Cadiz to see if they slipped up, but Nicholas Jackson's double on Wednesday ensured Real Sociedad still have work to do.
They have earned 17 points from the last 21 available and if they beat Atletico, would even have the chance to move third -- their highest finish since 2003.
Player to watch: Joselu
Espanyol are fighting for their lives at the bottom of the table and visit Valencia on Sunday, desperately needing a win. Spain international striker Joselu netted in the midweek comeback against Atletico and is their main goal threat.
Key stats
3 - Elche have not been defeated in any of their three games since being relegated
7 - After letting in only 11 goals in their first 33 games, champions Barcelona have shipped seven in their last three
8 - Red-hot Villarreal forward Nicholas Jackson has scored eight times in his last six La Liga matches
Fixtures
Saturday (all times GMT)
Sevilla v Real Madrid (1700)
Sunday
Valencia v Espanyol (1700), Cadiz v Celta Vigo (1700), Athletic Bilbao v Elche (1700), Girona v Real Betis (1700), Getafe v Osasuna (1700), Barcelona v Real Mallorca (1700), Atletico Madrid v Real Sociedad (1700), Almeria v Real Valladolid (1700), Rayo Vallecano v Villarreal (1700)
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