The website had reported that the 80-year-old could be suspended after the Saudi Arabian Grand Prix in Jeddah as part of the fallout from an investigation into the conduct of team boss Christian Horner.
Horner was cleared last week of accusations of misconduct, which he denied, by a female employee but has faced a call from Verstappen's father Jos to leave or risk the team being torn apart.
Marko told Sky Germany in Jeddah on Saturday that he had spoken to Oliver Mintzlaff, chief executive of corporate projects at the team's Austrian energy drink parent company Red Bull GmbH.
"It was a very good conversation. Of course there has to be calm back in the team. That's a priority. We agreed on all points. I'll continue here," said Marko.
"The term of my contract is still three years away. But as I said: there has to be calm."
The team's triple world champion Verstappen told reporters on Friday that Marko, who was instrumental in bringing him into the sport, had to stay.
He has been less emphatic about Horner, telling reporters in Jeddah that he and his father were very close and Verstappen senior was not a liar.
"It was very impressive. I'm very grateful to him for that," said Marko of the champion's comments. "But he is one of the few who has strength of character and shows loyalty."
Asked by Sky just before the start of the race whether Marko's staying was good news for him, Verstappen replied: "absolutely".
Mexican team mate Sergio Perez has made his own loyalties equally clear.
"Christian is a key member of our organisation, he is a massive team leader, he has put this thing together," he said. "The whole team is very united, really strong and we are all backing Christian and we just want to get on with our racing."
