This week, Simon won a stage and placed second overall in Paris-Nice and Adam won a stage in Tirreno-Adriatico on Sunday. Mitchelton-Scott will send Simon to the Giro d'Italia in May and Adam to the Tour de France in July.
"It's a very competitive market to arrive on a podium in a Grand Tour, not too many guys have done it, and you look at the usual suspects who are very consistent, but they are getting close," White told Cycling Weekly.
"They are progressing as we would've expected, and that's in the right direction."
Adam already placed fourth in the 2016 Tour and Simon seventh last year. Though the British 25-year-olds can read races and ride tactically smart, they continue to suffer in time trials and must improve to win or podium in the big three-week races.
"They are getting older, more experienced and stronger, it's normal for guys their age. They showed that they can match the best climbers in the world," White added.
"This year, you can see a difference. Look at Paris-Nice, the time trial that Simon did there was a great time trial. The only difference is that we don't get many time trials in a season. Simon's next time trial will be the Giro because there's no time trial in Cataluyna. That's why you have to nail the time trials when you get the opportunity.
"All three of our little guys [with Colombian Esteban Chaves] have been working on their time trialling, they are pure climbers, great climbers, but if you want to win stage races and Grand Tours, you need to be able to time trial.
"They have been working on it over the winter and they will continue to over their careers. If you aren't a natural time trial rider then you need to work at it."
Simon Yates finished 11th behind winner Wout Poels (Sky) in Paris-Nice's time trial, losing 33 seconds over 18.4 kilometres. Adam will have his chance to time trial on Tuesday in the final day of Tirreno-Adriatico.
From here, they both race the Volta a Cataluyna and then return to an altitude camp. Simon will start the Giro – with a time trial – on May 4 in Jerusalem.
"They need to just keep doing what they are doing. We know they are capable of winning stages, it's not new to them, every race they go to now we know that they are capable of challenging for a stage win and for the GC."
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