Madrid and Alonso went into their midweek Champions League game under intense scrutiny after a 2-0 home defeat to Celta Vigo on Sunday, which left Alonso's team with one win in their last five LaLiga games.
"We know there's a lot of pressure here," Rodrygo said. "It's normal when things don't go your way on the field. After my goal, I went to hug Xabi to show the team's unity. I knew it was important to have that moment."
A source told ESPN that club executives had met to discuss the coach's future earlier this week.
There were high expectations when Madrid hired Alonso to succeed Carlo Ancelotti but six months since returning to the club he once played for, he is facing an uncertain future. Madrid are four points adrift of league leaders Barcelona heading into Sunday's game at Alaves.
"It's a difficult time," Rodrygo admitted. "It is for us and it is for him [Alonso], too. Things aren't working out and I wanted to show people that we are united with our coach. I know a lot of things are always said. They often try to create a lot of things and I just wanted to say that, that we were united and that we need this unity to move forward and achieve our goal."
Rodrygo said he understood whistles from the Bernabéu crowd after back-to-back defeats.
"It's normal," the Brazil international said. "It's fair. We know that Madrid is a very demanding club, the fans are very demanding. We've been here for many years and we know that's what happens if we're not doing well. We have to try to improve and try to get out of this situation."
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At least Rodrygo ended his 32-game goal drought on Wednesday.
"I really needed the goal," he said. "I always try to score, always help, and the truth is, I wasn't at my best. These things happen in football. I have to stay focused and to keep training. That's what I've been doing all this time, knowing things weren't going my way.
"I'm sad about the defeat, but I hope that with this goal and this performance I'll get back to my best and help the team."
