A small number of the fans of the Dutch club who attended the Champions League tie between the sides in Glasgow two weeks ago ripped up seats and tried to barge past police and stewards to reach Celtic supporters, prompting fears of clashes next week.
Lawwell believes, however, measures have been taken to limit the potential for any disturbances, although he does expect a large number of Celtic fans to make the short journey to Amsterdam. There will certainly be more Celtic supporters in the city than those who can attend the fixture, but the chief executive is confident that violence will not flare.
"Our fans travel really well, there's a lot going and we have taken all the precautions, because the relationship is so good with Ajax, to make sure that it goes smoothly," Lawwell said. "No matter where you go, there's always that bit in the back of your mind that you want your fans to be safe, but the precautions we're taking will hopefully make sure that happens here as well."
Celtic sent Ajax a bill of £25,000 for the damage caused to Celtic Park during the Champions League tie last month. Seats were ripped up and thrown at police and stewards, although it was only a small proportion of the entire visiting support who sparked the trouble. Ajax intend to identify the culprits and force them to pay for the disruption that they caused.
"It was a minority out of 2800 people here who unfortunately behaved in a manner that nobody would like," Lawwell said. "It was no reflection on the Ajax directors or their management, who were first class. We accept that it's part of the game unfortunately that you'll get this minority and deal with it in that manner."
Midfielder Scott Brown will be in Nyon today to plead his case to have his suspension for being sent off against Barcelona on matchday two to be reduced. The Celtic captain received an automatic one-game ban, but two further matches were added following the incident with Neymar, when Brown kicked out at the Brazilian attacker. If he is successful, the Scotland internationalist could return for the game in Amsterdam.