According to reports, the BBC has made a number of offers to hire the Capital FM morning presenter, who is contracted to remain at Capital at least until the end of the year after signing a new contract in July.
The loss of Tarrant would be a huge loss to Capital FM. The station saw its morning show increase its audience share to 12.6% of the market in the most recent set of radio listenership figures.
The BBC is understood to be prepared to dig deep to woo the star to take over Sir Jimmy's popular lunchtime show as the 80-year-old presenter, who has resisted a sustained campaign to oust him, is moved to a weekend slot.
According to reports in the Evening Standard, sources say BBC chiefs have called Tarrant's agent seven times with six-figure salary offers. This would easily make the 56-year-old Tarrant the most highly paid DJ at the public service broadcaster. It is understood Tarrant earns £3.5m from his Capital and Millionaire? work.
Late last year, Radio 5 presenter Nicky Campbell claimed he had been offered the job. Other names in the frame for the BBC Radio 2 job are the BBC Newsnight presenter Jeremy Vine, GMTV presenter Eamonn Holmes and Radio 5 sports presenter John Inverdale.
Tarrant is by far the biggest name linked to the show and the offer comes as the popularity of ITV's Who Wants to be a Millionaire?, which has been on air for three years now, is beginning to wane as viewing figures fall dramatically. The latest series has lost 5m viewers on the previous series.
If he does go, he will lose millions in share options. However, Capital is likely to fight to keep him and have an option to keep Tarrant on for another year.
If you have an opinion on this or any other issue raised on Brand
Republic, join the debate in the .