Les Dogues took the streak to an unbeaten 1035 minutes on Tuesday night thanks to a 1-0 win over Marseille, and require clean sheets in their next two matches to break Bordeaux keeper Gaetan Huard's record of 1,176 minutes, set in the 1992-93 season.
"What can I say? I'm just enjoying the moment," Enyeama told beIN Sport "It's not just me though, it's the whole group. We stay strong like a team. We have the spirit and the strength.
"I don't know what is possible now. All I can say is that I have a great team around me and a great set of coaches."
Lille are now unbeaten in their last 11 matches, with Enyeama last conceding against Nice on September 15.