The experienced midfielder has taken the time to praise the former Tottenham man and is keen to make an impact again following his return from injury.
The Wales international impressed in Madrid's 7-3 win over Sevilla on Wednesday as he netted twice and created two more after struggling in previous games for the Santiago Bernabeu side.
Alonso is confident there is more to come from the 24-year-old.
"Bale's quality is clear for all of us to see and he will only further improve once he's settled in completely," the holding midfielder was quoted as saying on the official Madrid website. "He needs a period of adjustment to La Liga because it is very different to the Premier League. Plus we also need time to get used to him and vice-versa."
Alonso played his first game since his return from injury against los Rojiblancos and he was delighted to be back in action, where he is hoping get rid of any early-season rustiness.
"I'm very happy. I have no problems and I hope to slowly get there to be able to help the team," he said. "I feel like the team is taking shape but there are still things to polish. By reinforcing what we do well and correcting what we can do better, we hope to go as far as we want to."
The Spain international also briefly discussed Cristiano Ronaldo's celebration after his first goal as the attacker saluted the crowd in the wake of FIFA president Sepp Blatter's recent comments that the 28-year-old acts like a commander.
Alonso added: "He is right to take the situation with humor."