Ricciardo, 24, went on to his first career victory in Montreal, and the 2015 deal could be announced at Red Bull's home race in Austria next weekend, Sport Bild said.
According to the Australian Financial Review, the Perth-born driver has already hit the big time.
The publication estimated Ricciardo's base salary at $3.65 million this year, with a victory bonus of between $1m and $2m per win.
Bach and Garloff also claim that Red Bull's "long-term goal" is to keep Ricciardo alongside reigning quadruple world champion Sebastian Vettel until at least 2018.
That plan, however, could hit a snag.
German Vettel is struggling at the wheel of the RB10 this year, with Dr Helmut Marko admitting that, after four consecutive title triumphs, staring at the rear wings of the ultra-dominant Mercedes cars is "not exactly motivating".
Vettel's frustration began to show in Canada.
German reports quote him as saying after finishing the Canadian grand prix third: "I'm sorry to say it, but our cucumber simply doesn't go anywhere on the straights."
He also seemed to criticise Red Bull's strategy plan on Sunday, after he got stuck behind the Mercedes-powered Force India of Nico Hulkenberg in Canada.
"I was waiting for some smart strategic move on the part of the team, but it never came," said Vettel.
Vettel may also be alarmed with the impending semi-retirement of Red Bull's genius technical chief Adrian Newey.
When asked about Newey's plans, Vettel referred to his existing contract before adding: "I'm looking short-term, if you ask me about the future right now."
Newey reportedly tried to ease Vettel's fears, telling Auto Bild Motorsport: "If my colleagues need advice and help, I will always be there for them. Sebastian as well.
"He is not losing me."
Marko excused Vettel for any signs of mounting frustration, including the description of the RB10 as a "cucumber".
"If we do not give Seb a car capable of winning, we cannot be angry with him for being disappointed," he told Bild newspaper.
Undoubtedly, just as Red Bull was keen to keep Newey out of the clutches of high-paying rivals, the team may also be making plans to keep its multiple champion.
Auto Bild Motorsport said one 'project' of the forthcoming Newey-led 'Advanced Technologies Centre' could be a bespoke F1 engine, perhaps involving the well-known former Mercedes designer Mario Illien.
"It would not be the worst idea," Newey is quoted as saying.