The pair clashed during the final weeks of the leadership campaign, with Jenrick accusing Badenoch of being “disrespectful” to Tory members by failing to set out policies.
Badenoch said she would not have used that word and, defending her own approach, said she would not make a promise “unless I know how I am going to deliver it”.
However, following the conclusion of the contest the two candidates exchanged warm words.
In her victory speech on Saturday, Badenoch praised Jenrick’s “energy and determination”, adding: “You and I know that we don’t actually disagree on very much, and I have no doubt that you have a key role to play in our party for many years to come.”
Jenrick in turn congratulated his opponent on her win and urged Conservatives to unite behind the new leader.
On Monday, a friend of Jenrick’s said: “Rob thinks the party needs to come together and take the fight to Labour.
“Unity could not be more important. He’s eager to expose Labour’s dreadful record on law and order.”
A formal announcement of Badenoch’s full shadow cabinet is expected before its first meeting on Tuesday.
Former Chief Secretary to the Treasury Laura Trott has been given the job of shadow education secretary, with Neil O’Brien appointed shadow education minister.
Trott and O’Brien have already appeared in their new roles in the House of Commons at Education Questions.
Nigel Huddleston and Lord Dominic Johnson have also been appointed as joint chairmen of the Conservative party, the BBC understands.
It follows the appointment of Castle Point MP Dame Rebecca Harris as Tory chief whip on Sunday evening.