Norris lapped with a best time of 01:30.727 seconds, 0.058 faster than Leclerc in a reversal of the first daylight session, and 0.629 clear of last year's winner for Ferrari Carlos Sainz.
"The pace is good. I'm feeling good. It was a nice lap. I think we are doing what we expect, to be up at the front with Ferrari," said Norris.
"I was hoping to have a much bigger gap than I had, which means Ferrari are very, very fast. But things are going well so far."
Verstappen, 59 points ahead of closest rival Norris with seven rounds remaining, was 1.294 seconds off the pace with teammate Sergio Perez eighth at the winding Marina Bay street circuit.
Perez said the car lacked balance and it was "not looking great at the moment", with plenty more work to be done overnight. Verstappen agreed it had been a difficult day "not having the grip that we would like".
Mercedes driver George Russell, who ended the night seventh fastest, crashed into the wall in the closing minutes at turn eight but managed to reverse out and return to the pits minus the car's front wing.
"It was a really challenging Friday for us, we were really off the pace," he said. "We need to understand why that is.
"We made some big changes from FP1 into FP2 but the car is not really feeling as well-connected as it was this time 12 months ago and in recent races. So we need to try and get to the bottom of it.
"There's a lot of surprises out there. You've got the VCARBs (RBs) really quick, the Williams really quick, the Red Bull would seem to be off the pace and there seems to be a big gap to the McLaren and the Ferraris."
In a less serious incident, Argentine rookie Franco Colapinto pulled up at the Alpine pits instead of Williams and was waved onwards. He was 16th fastest with teammate Alex Albon ninth.
RB's Yuki Tsunoda was fourth fastest, ahead of McLaren's Oscar Piastri and RB's Daniel Ricciardo, whose future at the team is uncertain with Singapore possibly his last race, in sixth place.
Seven times world champion Lewis Hamilton, a four-time winner in Singapore, was 11th for Mercedes with Aston Martin's Fernando Alonso 12th.
Leclerc had set the fastest time of 1:31.763 in the incident-free first session, 0.076 quicker than Norris.
The pair are the only drivers to have started on pole position in the last five races and Saturday's qualifying will be crucial for Singapore, the only race Red Bull lost last season with overtaking always difficult.
Once-dominant Verstappen has now gone seven races without a win and looks likely to extend the run to eight on Sunday.
