Rogers is currently enjoying downtime after representing England at the tournament, having signed from Aston Villa for £117 million to become the most expensive British transfer in history.
Speaking for the first time since hijacking Arsenal's interest to secure the winger, Alonso said: "All the ones who come later because they reached the final stages of the World Cup, they will be ready.
"We have a few of them so they have to be ready. It's a great signing. In that position we needed an important player and I'm sure there were not many better options than Morgan Rogers. We were looking at him for a long time. He's going to be a key player for the team. Luckily it happened quickly and now we have Morgan.
"You need players who can have that almost instant impact, and I'm sure Morgan won't need much time to adapt to the club, to the system, to his team-mates. That was the idea that we wanted and to get a top player and Morgan is one of them."
Alonso was speaking from Sydney, where Chelsea are staging two pre-season friendlies against Western Sydney Wanderers and Tottenham Hotspur.
Asked about Crystal Palace defender Maxence Lacroix, who is expected to join for around £52 million in a deal believed to be imminent, Alonso said: "We want to build a competitive team. We are in that process. We have great quality, great talent, and we need to know what it takes to compete in the Premier League.
"We want a squad with the right balance between quality and mentality. Those things need to match. We have to be ready. The market is open for a few weeks. We need to be aware of different positions, different situations, so for sure, we have a plan, an idea, and we need to assess in the right moments. We can't say what's going to happen but some things might happen."
Once Lacroix is confirmed, he will become Chelsea's tenth senior centre-back, joining Levi Colwill, Trevoh Chalobah, Aaron Anselmino, Wesley Fofana, Tosin Adarabioyo, Mamadou Sarr, Josh Acheampong, Axel Disasi and Benoît Badiashile, underlining the scale of outgoing business the club's sporting leadership will need to manage.
"We need to know that probably there will still be some movements," Alonso said. "We need to be flexible and quick to act. The main thing is that we have a clear idea, a clear plan that we have built. Probably it will change until the last day of the market being open. I don't want to talk too early about our goals. We want to have a good season. We want to deserve to be as high as possible."
Chelsea covered the least distance of any Premier League club last season, a statistic Alonso is determined to address after his sides ranked among the league's most active while he was in charge of Bayer Leverkusen.
"We need to compete. We need to have good energy. We need to run. We need to work hard. We need to have the sacrifices which are compulsory and mandatory in each Premier League game. To win each game, it's tough. We need to prepare for that and we are using this time. We will see in the first game," he said.