“We’ve arrived at the beginning of the end of the thread (of investigations), and it relates to mafias, drug trafficking and weapons,” the official National News Agency quoted him as saying.
Mashnouq, who was speaking during a visit to France, did not elaborate further on any leads in the case.
His comments were the first official government comment on the investigation into the missing Czechs.
The five men disappeared in the Bekaa Valley of eastern Lebanon last Friday, along with a Lebanese citizen.
The cause of the abduction has remained a mystery, but there has been speculation it is tied to the case of a Lebanese man in Czech custody, Ali Taan Fayyad.
Fayyad’s brother was the Lebanese citizen who was went missing with the group, and his defense lawyer was among the Czech group.
Fayyad’s family has denied any involvement in the abduction, which was reported after the group’s car was found abandoned in the western part of the Bekaa.
Prague has declined to speculate on the case, or even confirm the group has been kidnapped.
Kidnappings of foreigners have been rare in Lebanon since its 1975-1990 civil war when some 100 foreigners, mostly Americans and West Europeans, were snatched.