McLaren leapt into a 20-point lead after Oscar Piastri's masterful victory at the Azerbaijan Grand Prix was compounded by Sergio Perez's late tangle with Carlos Sainz that ended with both in the wall.
McLaren's Lando Norris, who carved his way to fourth place after starting 15th on the grid, is hunting down the Dutchman Verstappen at the top of the drivers' standings.
The triple world champion's lead has been reduced to 59 points with seven grands prix and three sprints to go, leaving a maximum of 207 points up for grabs.
Verstappen won seven of the first 10 races this season but he has not triumphed in the last seven as McLaren, Ferrari and Mercedes have all taken race wins.
Red Bull's influential adviser Helmut Marko said after Baku that they have no chance of retaining the constructors' championship, but Verstappen has not conceded yet.
"We are going to work together as a team and the fight is not over," declared Verstappen after coming fifth.
"You win or lose as a team and we won't give up. It is as simple as that."
But history suggests that Verstappen's fortunes are unlikely to see an uptick around the unpredictable Marina Bay Circuit this weekend.
Red Bull have consistently struggled under the lights in Singapore.
It was the only race Red Bull failed to win last year, Ferrari's Sainz taking the chequered flag to deny Verstappen a record-extending 11th consecutive GP victory. Norris was second.
Verstappen has never won on the tight city-centre street circuit where tropical storms, intense humidity, concrete barriers, safety cars and red flags all add to the uncertainty.
Street circuit specialist Perez did win in Singapore in 2022, but that was Red Bull's only victory there since Sebastian Vettel in 2013.
McLaren milestone
After a strong start to the campaign, Red Bull are on the back foot.
"We have lost out on some significant points in the championship," team principal Christian Horner said after their messy Azerbaijan outing.
"However, we will brush ourselves down and fight back hard."
McLaren, by contrast, are in buoyant mood as they target a first team world title since 1998.
"First in the constructors' championship is an important milestone in our journey," said team principal Andrea Stella.
"However, the team remain clearly focused on the task ahead. We quickly turn our attention to Singapore."
The high downforce track in Singapore has traditionally been of benefit to Mercedes.
"We've also got lots of data to work through ahead of Singapore," said Lewis Hamilton, trying to take some positives from a ninth-place finish in Baku after an engine change forced him to start from the pit lane.
"We've got some upgrades coming before the end of the year so hopefully we can make a step closer to those at the front soon."
Ferrari's Charles Leclerc won at Monza but he could not make it two in a row after starting from pole position for the fourth year in a row at Baku and, as on the three previous occasions, failing to convert into a victory.
"For sure we have a bit of frustration because we had the space and pace to win this weekend," lamented Ferrari chief Fred Vasseur.
In the end Leclerc did well to hang on to second after his hard tyres degraded badly behind Piastri in the second half of the race and then saw teammate Sainz's late crash in his mirrors.
"It wasn't the best day for our team," said Leclerc, who is only 19 points behind Norris and still firmly in the hunt for the drivers' title.
"But we now head to Singapore and will come back stronger."
