There was no immediate comment from either team on Thursday but senior insiders confirmed the move, first reported in Dutch media, to Reuters.
The news was also reported by the BBC and Sky Sports, with 2028 given as the likely start date for a man who has been working with Verstappen since 2016 and has played a key role in helping the Dutch driver to four world championships.
Lambiase, 45, had also been linked with Silverstone-based Aston Martin, whose team principal is former Red Bull star designer Adrian Newey.
While Aston Martin have endured a nightmare start to the season, struggling to even finish races with an uncompetitive Honda engine, McLaren won both titles last year with champion Lando Norris and teammate Oscar Piastri.
Lambiase is expected to become head of race engineering at McLaren once a potentially long period of 'gardening leave' comes to an end, with former Ferrari engineer Stella continuing in his position.
Stella, who worked with Michael Schumacher in a golden era at Ferrari in the early 2000s, has a multi-year contract with McLaren and no intention of returning to Maranello despite some media speculation about his future.
The close but forthright relationship between Verstappen and 'GP' over the team radio has become a familiar part of Formula 1, similar to the pairing of Lewis Hamilton and Peter 'Bono' Bonnington during the seven-times world champion's spell at Mercedes.
Former Red Bull boss Christian Horner, fired last July, once compared the relationship to that of "an old married couple arguing about what to watch on television: "The dynamic between the two is so intense that in between you have to ask yourself who is supposed to be the driver and who is supposed to be the engineer here."
Losing the Briton will be a blow to Verstappen, after the departure of other important figures in recent seasons and once-dominant Red Bull's waning performance on track, but the 28-year-old has also increasingly cast doubt on his own longevity in the sport.
“I'm thinking about everything inside this paddock,” he said in Japan last month. Verstappen is no fan of the sport's new engine era and rules that force drivers to manage energy deployment and take corners at less than full speed.
In 2021, when they won a first title together, the Dutchman went so far as to say that he would not continue without Lambiase.
"I have said to him I only work with him. As soon as he stops, I stop too," he told Dutch broadcaster Ziggo Sport.
"We can be pretty strict with each other sometimes but I want that. He has to tell me when I'm being a jerk and I have to tell him."
McLaren already have former Red Bull employees Rob Marshall and Will Courtenay in senior roles as chief designer and sporting director respectively.
