Red Bull insists it is "very happy" with the Australian driver, but his contract expires this year and Lewis Hamilton has been flirting with a switch from McLaren.
Renault team boss Eric Boullier admits he has spoken with Webber, 34, about moving to Renault, whose undisputed number one driver Robert Kubica is currently recovering from horror injuries sustained in a February rally crash.
"I really like Mark and have known him for a long time," the Frenchman said in Valencia.
"We have discussed it from time to time but we have never pushed him to ask what he wants to do because that decision is up to him."
Webber is managed by Flavio Briatore, Renault's former long-time team boss.