The Spaniards were well beaten by modest Portuguese side Braga, who will debut in Europe's top club competition after a surprise 5-3 aggregate win.
Tuesday's 4-3 second-leg defeat at their Sanchez Pizjuan stadium saw Sevilla booed off the pitch, and president Jose Maria del Nido struggled to hide his disappointment.
"There's no need to consider making changes, at the moment I think it would be premature and it would be bad for the entity," Del Nido told Spanish television when asked about Alvarez.
Coming hot on the heels of Saturday's 4-0 drubbing against Barcelona in the Spanish Super Cup, Del Nido added, "We haven't achieved either of our two objectives for August. (But) football continues and we still have the Europa League."
Alvarez, who replaced Manolo Jimenez in March, has struggled to win over fans despite leading them victory in the King's Cup final in May.
He told reporters he had no intentions of quitting.
"The season's only just starting, we have received two hefty blows but I'm only thinking about training again tomorrow, and lifting the morale of my players" he said.
Sevilla kick off their league campaign away to promoted Levante on Saturday.
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