The BBC is said to be so desperate to keep them, following the sacking of Jeremy Clarkson, that they are prepared to fork out £4.6m to secure them.
Richard and James have been offered a new 'Top Gear' deal
According to The Times, the new deal would tie them in for a further two or three series of the hit motoring show.
It is thought the offer would see them take home between £760,000 to £1.15m each per year - an increase of the reported £500,000 per series they were previously earning.
News of the deal comes after BBC boss Danny Cohen admitted he didn’t want to keep splashing money at the expense of making less programmes.
"If the BBC takes on more financial obligations we've got less money to spend on content. It's as simple as that,” he said at a Media Summit Q&A earlier this week.
"If we took on one of those things, we'd just make less programmes. And that's something I think none of us want to do."
The trio has been broken up after Jeremy's sacking
However, both James and Richard have said they will not carry on with ‘Top Gear’ without Jeremy.
“Me and [Richard] Hammond with a surrogate Jeremy is a non-starter, it just wouldn’t work. That would be lame, or ‘awks’ as young people say,” James told The Guardian earlier this year.
“It has to be the three of us. You can’t just put a surrogate Jeremy in and expect it to carry on. It would be forced. I don’t believe they would be stupid enough to try that.”
Richard added on Twitter: “There's nothing for me to 'quit'. Not about to quit my mates anyway.”
Philip Glenister, Jodie Kidd and Guy Martin have all been touted as possible replacements for Jeremy, while the likes of David Hasselhoff have also said they would like to take on the role.