Scottish star Craig Ferguson, who has been in the hot seat on the programme since 2005, announced in April that he would be stepping down from his post by the end of the year.
Now actor Corden - who stepped down as the host of the Brit Awards earlier this year - is said to be being lined up to replace him, according to US news website The Wrap.
His potential appointment has not been confirmed by either the network CBS or Corden's representatives.
Corden, 35, was seen on BBC1 earlier this year interviewing Gary Barlow for a documentary, and he also fronts the Sky sporting panel show A League Of Their Own, which will run for at least another three years under a new deal.
The co-writer and star of Gavin And Stacey is currently appearing alongside Keira Knightley and Mark Ruffalo in the film Begin Again, and will soon be seen in the cast of a big screen version of Broadway musical Into The Woods, which also stars Meryl Streep and Johnny Depp.
Corden's US profile was boosted when he won a Tony Award in 2012 for best leading actor in a play for his starring role in One Man Two Guvnors, which transferred to New York from the West End.
Ferguson's decision to exit the series came in the wake of another chat host David Letterman announcing he was retiring from his programme The Late Show.