Christos Tzolis, signed for £34m this summer, remains Arsenal's only major addition to date, though that is expected to change soon: Bruno Guimaraes is due at the club for a medical on Monday, and talks continue over a move for Real Madrid's Vinicius Jr, whose deal in Spain runs out next summer.
Arsenal ended a 22-year wait for the league title last season and were desperately close to a historic double, only for Havertz's goal in the Champions League final to prove insufficient as PSG won the shootout in Budapest.
It is against that backdrop — and with Chelsea, Manchester City, Manchester United and Liverpool all having made significant additions of their own this summer — that Havertz is urging his club not to stand still.
"I think we can and have to go further. You see other teams getting better so we have to step up another level again," he said.
"That's what we are all here for, that's what we train for, play those test (pre-season) games and reach our goals this season again."
Having finished as runners-up in three consecutive Premier League seasons before finally getting over the line last term, Havertz made clear the club's ambitions now extend well beyond simply retaining the title.
"I think we want all the trophies," he said. "That's what we're here for. We missed out on a lot the last couple of seasons. Obviously, we won the Premier League now, but there's a lot more that we can do.
"We have to reach another level to make that happen. We know how long the season is, we know how hard it is to go through the competitions. We want to do better again this year and make things happen again."