Verstappen got the edge over Leclerc in Austria with a fierce overtake, taking Red Bull’s first victory of the season.
But Leclerc came back with his own aggressiveness against the Dutchman at Silverstone, overtaking him and aggressively defending his position on the track.
The Ferrari driver is looking forward to what the future rivalry will hold after racing against each other when karting.
Leclerc said to BBC Sport: "I really hope so.
"We had some very good fights in karting that didn't always end in a good manner, but it was fun good times.
"And I can't wait to have more with him.
"In the end, we have done most of our career together, so it is fun to find ourselves again here in Formula 1."
The Monegasque is hopeful that his first win in F1 comes soon after having been so close in Bahrain had it not been for an engine failure.
"Well, hopefully, the first win to come very soon," he added.
"And many more after. But at the end the target is the first win now, trying to take the opportunity whenever we have it.
"In the future... my dream as a child was being a world champion.
“Now I am finally in a team where I think this can be a reality so we need to keep working and hopefully this will happen one day."