Ex-Formula One driver Brundle is a "key target" for Sky as the broadcaster prepares to put together a team to cover every race, qualifying and practice session live between 2012 and 2018, according to The Mirror.
From next year, the BBC will air only the "key races", including the Monaco and British Grands Prix, on free-to-air television, while the corporation will carry highlights of any races that it does not show in full.
Brundle said during the Hungarian Grand Prix at the weekend: "I am still trying to understand the exact details of the deal and cannot make any decisions until then."
When the agreement was announced last Friday, he tweeted: "Found out last night, no idea how it will work yet - I'm out of contract, will calmly work through options. Not impressed."
The decision to make half of all Formula One races only available on pay-TV has been met with outrage from fans angry at the prospect of having to pay a monthly Sky subscription to watch the entire championship live.
A protest petition has been launched on the Petition Buzz website calling for coverage of all Formula One races to remain available on free-to-air television. As of this afternoon, the petition has attracted 22,664 signatures towards the aim of compiling 1m to heap pressure on the BBC.
However, the BBC's Formula One presenter Jake Humphrey said in a blog post today that there was a "very real threat that F1 was going to be lost from the BBC for good after this year".
The new deal is expected to save the BBC around £30m a year, and the corporation claims that it would have had to drop Formula One entirely unless it had agreed to share the rights.
Humphrey said: "I, like many of you, had seen the headlines over the past month and I suspected that we would get some news sooner rather than later.
"However, contrary to popular belief I didn't have the inside track on what was happening and on Thursday evening I went for dinner and then bed thinking it was set to be just like any other race weekend. I then woke up at 7am as my phone was ringing - Ben Gallop the BBC F1 boss had flown to Budapest, was in the lobby of the hotel and had news on the TV deal.
"The next few hours went by in something of a haze. There was some understandable sadness and confusion as the news sunk in that things were changing. I think my breakfast consisted of one mouthful of coffee."
Humphrey praised the "talented and dedicated team" behind the BBC's Formula One coverage, including analysts Eddie Jordan and David Coulthard, but accepted that things will be different from next season.
"We will be working even harder to make the end of this year a resounding success on the BBC and looking to be stronger, better and more professional than ever," he said. "There's no disguising that from next year it will be different. I know it is frustrating for you to not yet know the finer detail - and as soon as we have it worked out, you will be the first to know.
"From the presenting team to the races we cover live, to the transmission times of the highlights shows and how we make those a success too. You will again be central to how the BBC make those decisions.
"And after 2011 please be assured of this: the BBC will endeavour to make the overall F1 offering as enthralling and compelling as possible. We never take our work for granted and we will be always be driven by that desire to produce the best shows for you, our audience."