Trump rips Obama hours before summit with Putin
Report: Trump administration officials urged furious Tillerson not to quit
Pence reportedly asked Tillerson to remain at least until the end of the year.
Tensions reportedly came to a head in late July after Trump delivered a highly controversial speech to the Boy Scouts of America, which was once led by Tillerson.
The secretary of State also referred to the president as a "moron" a few days earlier during a meeting with key Cabinet officials, according to NBC.
State Department spokesman R.C. Hammond disputed the report, telling the network that Tillerson never called Trump a "moron," and never considered quitting this summer.
This would not be the first time Trump and Tillerson appeared to have clashed.
When asked to comment about Trump's remarks on the racially charged violence in Charlottesville, Va., in August, the secretary of State told Fox News's Chris Wallace that the president "speaks for himself."
Trump in a series of tweets on Sunday also appeared to jab at Tillerson for looking to engage in direct talks with his North Korean counterparts.