England finished fifth in the tournament after losing their last three matches, leading to criticism of Australian coach Jones.
RFU chief executive Steve Brown said while England's campaign was "very disappointing", Jones retains the "unwavering support" of the governing body.
"At the moment there is no knee-jerk reaction," Brown told BBC Sport. "The key now is how we respond and turn things around.
"The reaction I've seen from Eddie and the coaching staff gives me some confidence that we are in a good place and we will get this back on track.
"We are on the right trajectory, although clearly not in the last three games. The reaction for me is the test as to whether someone has the calibre and character, and I've seen that.
"We will never get a bigger test than South Africa. It's as big a series as there is ever going to be," he added, referring to the tour against the Springboks in June.
Jones apologised after being caught on camera calling the Irish "scummy" and Wales a "little s*** place", with the RFU ready to move on from the incident.
"Eddie has apologised to me personally," Brown added. "We have drawn a line under the matter. It doesn't fit with the way we think or the way Eddie thinks."
England's tour of South Africa gets under way on 10 June (AEDT).