According to a source familiar with the deliberations, Christie wants to run and has been discussing the idea with the Trump transition team for several days.
The team believes the role is ideal for Christie because it puts him in a high-profile position and does not require Senate confirmation.
ABC News reached out to Christie and the RNC for comment but did not immediately hear back.
Reince Priebus, the current RNC chairman, has been chosen to be Trump's chief of staff.
The RNC chair is elected by the committee's 168 members. If Christie is seen as the clear choice of Trump, that will likely clear the field of any challengers.
Christie has been one of Trump's most prominent supporters since ending his own presidential campaign earlier this year. Trump announced in May that Christie would lead his White House transition team but replaced him after Election Day with Vice President–elect Mike Pence.
Christie served as the chairman of the Republican Governors Association from 2013 to 2014.
Earlier this week, he said he wanted to remain as governor through the end of his term. "I am completing my term," he said at a news conference Tuesday. "Now, I said this on the radio last week, and some people said it was equivocal because I said, if a president of the U.S. comes and asks you to do something, you don't say no before they even ask. I am telling you I am completing my term.
"Now, I will also tell you that if something extraordinary happens in the world where my service is needed, I will consider any requests that are made," he added.
ABC News' Jordyn Phelps and Shushannah Walshe contributed to this report.