Jones was sacked on Tuesday after overseeing England's worst year of results since 2008.
The former England captain was given his first coaching role in 2012 when Jones was in charge of Japan and the two joined forces again in 2015 as Borthwick became England's forwards coach.
He stepped down from that role following the 2019 World Cup final defeat to South Africa.
The 43-year-old then moved to Leicester, who he has transformed from fallen giants to last season's English Premiership champions.
Leicester begin their European Champions Cup campaign against Ospreys on Sunday and Borthwick said he will be in the dugout.
"I don't think anyone can possibly doubt how committed I have been since the minute I walked through this door to give the best I can for this team," Borthwick said at his pre-match press conference on Friday.
"That is my intention again this week to get the performance we want."
However, he did take the chance to pay tribute to Jones' seven-year reign as England coach.
"Eddie is a phenomenal coach and I was very fortunate and privileged to have worked with him for so many years," added Borthwick.
"He never ceases to want to be better and improve the players to help them achieve their dreams.
"But what I should actually talk about is the generosity of the guy. Whenever I've heard a coach ask questions of him I've seen him help people all around the world.
"He loves the game and he's just an incredible coach."
