But the energy drinks manufacturer is keen to keep the driver, who they have backed since 2001, within their driver portfolio.
Red Bull has completed a deal with PKV to sponsor their cars next year in return for one of their own drivers racing for the team.
Red Bull motorsport advisor Helmut Marko confirmed that Klien has 10 days to make up his mind, and will be dropped from the company's books if he says no.
If that happens, Swiss Toro Rosso test driver Neel Jani will get the drive instead.
PKV are likely to keep current drivers Oriol Servia and Katherine Legge next year in what is set to become a full-time three-car lineup, despite Jimmy Vasser's decision to retire completely.
Another driver linked to the third seat is Briton Dario Franchitti, who has not yet been offered a new deal with IndyCar squad AGR and could return to the series where he won 10 times from 1997-2002.
Franchitti could even be in the running for a drive with champions Newman/Haas Racing if double champion Sebastien Bourdais makes an unlikely switch to Formula One as BMW Sauber's test driver.
Further confusing matters is an ongoing doubt as to whether 2002 series champion Cristiano da Matta will return to racing next year.
Brazilian da Matta collided with a deer during a Road America test session earlier this month and is still recovering from serious head injuries in hospital.
His RuSport drive has been left vacant for the past two races, but will almost certainly be filled if he cannot return next year.
Rocketsports Racing owner Paul Gentilozzi tried to establish a Team Brazil at the start of the season, and with Amazonian Antonio Pizzonia returning at Montreal last weekend, plans are back on.
Gentilozzi has earmarked ChampCar Atlantic frontrunner Rafael Matos as Pizzonia's potential team-mate next year and plans to test him soon.
Dale Coyne Racing look set to test Briton Jay Howard – who leads the Indy Pro Series - and Italian Enrico Toccacelo in the near future, although Toccacelo belives he could make his debut before the season's end.
Finally, both Czech Tomas Enge – who drove in IndyCar last year – and Chinese driver Ho-Pin Tung are talking to a number of teams about 2007.
Enge, 29, was actually on the verge of replacing Andrew Ranger at Conquest for the final three races of this season before the Canadian put together a late sponsorship deal to save his seat.