The 27-year-old France captain, who equalled Cristiano Ronaldo's club record of 59 goals in a calendar year for Real in their 2-0 LaLiga win over Sevilla on December 20th, has a sprain in his left knee.
"We'll definitely miss Kylian (Mbappe)," Alonso told reporters. "We have to have a lot of faith in those who are going to play tomorrow.
"Now we'll see. We have to wait and see how things develop, and we'll do everything we can to get him back as soon as possible.
"We're going to be pushing the deadlines; it's very much a matter of feelings. When is as soon as possible? That's the question. I don't know. The Super Cup? We hope."
Alonso's men were five points clear at the top before their form took a downturn in November.
They drew three league games in a row before suffering a shock loss to Celta Vigo at home in December, allowing defending champions Barcelona to surge four points ahead in the title race.
Real's dip in form has coincided with a goalless run for forward Vinicius Jr. The 25-year-old Brazilian winger, who scored 22 goals last season, has struck just five times across all competitions this term and has not found the net since October.
When asked how Vinicius is doing after being booed by a section of fans in their last match against Sevilla at the Bernabeu, Alonso said: "Vini looks fine to me, cheerful and smiling. The break has been good for all of us to recharge our batteries.
"Tomorrow, we are the ones who have to give in order to receive, to spread that enthusiasm and rhythm that we need. If that flows in both directions, we will enjoy ourselves."