Four-time world champion Verstappen's race engineer Gianpiero Lambiase has already signed a deal to join McLaren when his contract allows, by 2028 at the latest, and the Dutch driver has been unhappy with Red Bull's performance.
Verstappen is known to have a break clause in his contract and continues to be linked in media speculation to McLaren, despite them saying they are happy with the current pairing of Piastri and world champion Lando Norris.
"I've got a contract in place, multiple reassurances that the team are very happy with me and I'm very happy with the team," Piastri said at last weekend's British Grand Prix.
Verstappen crashed out of that race and said his car's rear wing was "super-dangerous", with the incident further fuelling speculation about a rift within the team.
Webber told Racer.com that Piastri was contracted to McLaren "for the foreseeable future".
"Talk of him agitating to leave is nonsense," said the former F1 driver.
"There has been a lot of fiction written about him and other teams... McLaren have repeatedly said they want him for the long term and Oscar is focused on that."