Hamilton was fastest in the first session on a day when Ferrari's pace caught the eye of championship leaders McLaren.
McLaren and Red Bull are two of a number of teams who have a new floor for Silverstone, while Ferrari are continuing with the new design they debuted in Austria last weekend.
Hamilton - who has had a difficult start to his first year at Ferrari but has won more times at Silverstone than anyone else, with a record nine victories - said he was finally becoming more at home in the car.
"I'm progressing a lot with the car and much more comfortable on knowing where it needs to be. By P2 we still weren't where we needed to be so struggled a little bit more but definitely know the changes we need to make to the car."
Leclerc, who was the fastest driver on the race-simulation runs at the end of the second session, said: "The day was good. We've been pretty strong so far. It's positive.
"We still need to find some pace in qualifying. McLaren is once again probably the car to beat but in race pace I was happy, I am finding my way.
"I am changing quite a lot the car in weekends recently to try to find some pace in qualifying. For now I don't seem to find the way for that but in the race I am very happy with where we are.
"We are very strong in the race but we have to do a step forward in qualifying."
Norris, 15 points behind Piastri in the championship after his win in Austria on Sunday, said: "The Ferraris have been very, very quick today and they shall be tomorrow.
"We have a bit of work to do. It looked maybe a bit too good today. The Ferraris always catch up into P3 like they did last weekend, so pleased with today but nothing to be too proud about just yet."
Verstappen said: "It was quite a bad day, just no balance in the car and quite difficult corner to corner. So quite a poor day for us in general."