Together they have been enormous figures in Celtic's modern history. They each understand the pressure the other is under and they co-operate easily, even if an occasional element of tension is inevitable given that one of them needs to spend money and the other is committed to balancing the books. Maybe Lennon feels his chief executive could have done more to back him in the transfer market. Maybe Lawwell feels his manager should have more to show for the £10.2m he spent between June and August.
Both arguments are valid and they are not mutually exclusive. What unites both men today is the criticism coming at them after a pretty wretched six-match Champions League sequence.
What happened in Barcelona on Wednesday was hard for Celtic to take because they are not a club used to dealing with losing four, five or six goals. What they often do to others was, for once, done to them. The brilliance of Barcelona, and particularly the mercurial Neymar, meant there was no shame in being on the end of the punishment but still it was humbling to be reduced to fodder for what became a glorified Catalan training exercise.
The collapse at Camp Nou was symptomatic of the problems already evident in a failed, unimpressive campaign. The key statistics confirm that Celtic had little hope of surviving in this season's group, as was evident from the fact they lost five out of six games. Of the 32 teams in eight groups. only two had fewer goals, and fewer attempts on goal, than Celtic. Only two conceded more goals. Only two averaged less time in possession (41%). No other team made fewer passes.
That is a brutal stripping down of a side that does not make or convert enough chances, does not have enough of the ball, and concedes too easily. They played to par by being the fourth seeds who finished fourth, but only in the away defeat by AC Milan and the narrow home win over Ajax did they deliver persuasive performances.
Celtic have taken a significant step back in Europe, from 10 group points and a last-16 place in 2012/13 to three points and finishing bottom of the group this time. The last three performances were particularly alarming, the defeats by Ajax, Milan and Barcelona, because they were characterised by a flatness and timidity which left Celtic unrecognisable from the spirited, tenacious side which did so well last season.
The same set of players have just delivered two of the richest domestic performances of Lennon's career as manager, with 12 unanswered goals against Hearts and Motherwell, yet they have become lambs in Europe. The confidence and sense of belonging they had created as a Champions League side (and which was evident even when they lost in the San Siro) now seems lost.
The essence of the matter is that an impressive Celtic team was unavoidably broken up - Gary Hooper, Victor Wanyama and Kelvin Wilson all wanted to leave - and the replacements have been too poor to maintain the previous standard. Teemu Pukki, Derk Boerrigter and Amido Balde have been deeply disappointing. There has been no change in Celtic's signing policy during all of this: the six aforementioned players all cost between £1m-£3m. The crucial difference was that Hooper and Wanyama, in particular, were outstanding signings which improved the team hugely, while this season's recruits have been mediocre and contributed to its steep deterioration as a presence in the Champions League.
When a club signs players for around about £2m it cannot be sure of what it is getting. That fee is not a guarantee of quality at the level of European football Celtic wish to reach. Lawwell knows that there will be an intensification of the demands for the club to spend more, and to bring in a goalscorer worth perhaps £6m or more to virtually guarantee substantial improvement. That would be a move even Lennon has warned against in the past.
A player who costs £6m would have correspondingly high wage demands, probably expecting a far higher salary than any of the existing players. Jealousies and resentments spread easily in a dressing room. Bringing in one marquee name could burden the club with an entire round of wage rises to maintain harmony.
When Celtic have been prepared to commit to a more expensive signing they have run into another problem: the more a player is rated, the harder it will be to persuade him to come to the SPFL Premiership. They wanted the £4m-rated Artjoms Rudnevs from Hamburg in July but he was not interested in playing in Scotland. There was a similar message from Kostas Mitroglou, the Olympiakos and Greece striker, and another £4m man. Whenever Celtic identify a striker they want, they can be pretty sure that at least another 30-40 clubs could beat them to him if they want.
Celtic have moved into a strange no-man's land: out on their own as the dominant force in the Scottish top flight but struggling to keep their head above water as a club that can compete in, or even qualify for, the Champions League proper. In September they posted an annual pre-tax profit of nearly £10m and they have made more than £40m from back-to-back Champions League campaigns plus income from player sales. Doubtless that will soon include the departure of goalkeeper Fraser Forster for a near eight-figure fee, meaning the loss of another key figure from the vintage 2012 side.
They are cash rich but isolated, unable to commit to major buys because their long-term financial stability is dependent on annual Champions League participation which cannot be guaranteed. As Lennon said in a characteristically withering response to a poor performance at Camp Nou, Celtic need to rebuild if they are to climb back as a force in Europe. When he sits with Lawwell they will both expect more from each other: the manager wanting more resources, the chief executive more value for money. It is perfectly possible they will both be disappointed.