The 39-year-old’s anticipated move to Welford Road would not take place until after the World Cup in Japan, where he is working as England forwards coach.
The exact nature of Borthwick’s role with Leicester is not yet clear, nor is the impact his arrival would have on head coach Geordan Murphy.
Tigers have so far declined to comment.
Having Borthwick join the coaching team would be the latest in a series of backroom reshuffles in the past 13 months by the East Midlands club.
Murphy was promoted to head coach – initially on an interim basis – after Matt O’Connor was sacked just one game into the 2018-19 season.
Former England and Ireland coach Mike Ford was recruited later in the season to help Leicester avoid relegation and he has stayed on as attack coach.
Borthwick, who captained both Bath and Saracens during a 16-year playing career, has been part of Eddie Jones’ England backroom team since 2015, having previously worked as forwards coach at Bristol.
His move to again link up with Jones, having previously worked under the Australian at Saracens and with the Japan national team, caused controversy as Bristol initially blocked his departure.
A settlement was eventually agreed, and Borthwick has since added to his reputation as one of England’s brightest coaches.
Any move to Leicester would likely mean the club having to pay compensation, as he remains under contract with the Rugby Football Union.
Leicester, who were put up for sale in June, are looking for a major improvement on their second-from-bottom finish last season.
Despite that, they remain one of the most successful clubs of the professional era, winning the Premiership 10 times and the European Cup twice.
They began the current season with a heavy defeat by Worcester in the Premiership Rugby Cup, a result that was followed by a home win against Exeter Chiefs in the same competition.
Leicester’s league campaign, delayed to accommodate the World Cup in Japan, gets under way with a trip to Worcester on 19 October.
Chris Egerton, Leicester Tigers commentator for BBC Radio Leicester
Borthwick’s undoubted technical skills as a player and now as a coach make his impending arrival at Leicester quite a coup for a club seeking sharp improvement after last season.
Roles have not been explained, but a man interested in becoming England head coach at some stage would surely want to be in charge of first-team affairs.
The Tigers board have jumped at the chance to hire a man whose signature would be coveted by many other Premiership clubs.
But what about the timing? Since he became Leicester’s fourth head coach in 18 months, Geordan Murphy has overseen a restructure which involved the departure of over 20 players and promotion of the club’s successful Academy players.
After barely 12 months in the post, and repeated assurances in public that Leicester would give Murphy what he needed, the ex-Ireland international will feel he has not had the time to complete the job.