The 43-year-old Swede plans to leave the club he served as a player between 1998 and 2004 and assistant manager for the past four years.
Garry Parker could step up from his role as coach, with any new addition to the backroom staff stepping into his current role.
The work being done by John Kennedy and Stevie Frail at Development Squad level -where Lennon cut his teeth in coaching before getting the top job in 2010 - is hugely respected.
Head of Youth Development, Chris McCart, is also highly rated within the club.
However, it may be judged prudent to leave these coaches to continue the important work they are already doing, and the search will extend well beyond the perimeter of the Lennoxtown training complex.
Although Mjallby will stay on until the end of the season, with the squad due to report back on June 24, a pre-season training camp booked for Austria soon after, and the opening Champions League qualifiers taking place on July 16, Lennon can understand the reasons to begin the search for a replacement immediately.
The manager will also be moving in on the signing targets the scouting network has identified, and having another opinion on possible recruits would be an advantage.
However, because Lennon considers it a key position in his backroom team, he will not rush into any appointment.
The Hoops boss said: "When I was given the chance to be in charge of the team, Johan was the obvious choice to join me in tackling the challenges ahead.
"From day one, he has been a very important part of our success. Johan will always be a great friend. I'll be very sad to see him leave the club. He has given us such great service over a number of years and he will always be a big part of Celtic. I know whatever challenge Johan chooses next, he will be a great success."
Mjallby is expected to head home to Sweden while he considers the job offers presented to him.
AIK Stockholm, where he made his name before moving to Celtic in 1998 as a £1.2million signing for Dr Jo Venglos, will have been alerted to his availability.
Their sports director, Bjorn Wesstrom, said: “A person with such a natural link is always interesting to us.
“There’s still dialogue between us. We need to talk more to him first.
“It would be fantastic, but we’ve lost a lot of money in the last year and will need to see what we can offer.”
Mjallby is keeping his options open and said: “I’ve enormous affection for Celtic and it is with sadness that I now leave to seek some different opportunities.
"The club has been part of my life for almost two decades and I’ve been dedicated to them in every way, first of all as a player and then in management.
"We had some great moments and memories I will always cherish."
Chief executive Peter Lawwell will work closely with Lennon in the search for Mjallby's replacement. He expressed his appreciation for all that the Swede has done in his time at the club and said: "Johan has given Celtic so much."