Alonso's position as coach has faced further scrutiny after Madrid's 2-1 home Champions League defeat to Manchester City on Wednesday -- a game which Mbappé missed with a short-term injury.
ESPN has reported that club executives met earlier this week to discuss Alonso's future after a run of one win in five LaLiga matches, and the team were whistled by the Bernabéu crowd after the City loss, which came with eight first-team players unavailable.
"[Mbappé] is fine to play, we've got him back," Alonso said in a news conference on Sunday.
"We'll decide tomorrow. That's obviously good news."
Mbappé was later named in Madrid's matchday squad for their trip to Alavés, while centre-back Dean Huijsen also returned to the group, one of just three first-team defenders, alongside Antonio Rüdiger and Raúl Asencio.
As well as injured players such as Éder Militão, Trent Alexander-Arnold and Eduardo Camavinga, Madrid are also without the suspended Álvaro Carreras and Fran García after they were sent off in last weekend's home loss to Elche.
"Despite the absences, we have strengths and a squad large enough to field eleven Madrid players who will give their all," Alonso said. "The desire, the commitment to win will be there. It's an important match."
Madrid began the weekend four points behind rivals Barcelona at the top of the table.
"I've been in football for many years," Alonso said, when asked about reports that his sacking is imminent.
"Nothing that happens surprises me. These are normal things, things that have happened in the past, and will happen in the future.
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"We have to face them with the responsibility of the position and what we represent. From there, it's time to work to turn the situation around. It's not something that surprises me too much."
Alonso said he's had "constant communication" with president Florentino Pérez and other senior figures at the club in recent days, and praised reserve team coach and former teammate Álvaro Arbeloa, who ESPN has reported has been discussed as a possible successor.
"I think Arbeloa could be Madrid coach in the future," Alonso said. "He's doing things well."
