Lennon, who took over at Parkhead in June 2010, is on a rolling contract but has been in talks with the club over a new deal.
Brown and Lennon were in direct opposition as players at Hibs and Celtic respectively, before the Northern Irishman hung up his boots and moved into management.
Having started the defence of their Scottish title with a 2-1 win over Ross County, Celtic are in Sweden to defend a one-goal lead in the second leg of their Champions League tie against Elfsborg.
"The manager signing a new deal would be a boost for everybody," declared Brown.
"Our relationship has definitely changed since we played against each other. Back then it was a hate-hate relationship between us.
"He's great to work under. He wants us to get the ball down. I'm not saying we're Barca but he wants to get as close to that philosophy as possible.
"He joined in the training game the other day and still wants to play. He passes on everything he did in his career and helps us in every way."