The German daily Die Welt reported that VW had agreed to pay USD 5,000 (4,400 euros) to each owner in the US of a car fitted with the suspect software that skews the emissions of its diesel engine.
On top of this, VW would foot the bill for re-fitting each engine, the newspaper said.
Some 600,000 vehicles in the United States are involved.
The speculation sent VW's shares soaring 7.5 per cent to an intraday high of 130 euros on the Frankfurt stock market today, after they already jumped by more than six per cent the day before.
In early afternoon, the shares were showing a gain of 5.6 percent in a slightly softer overall market.
Neither VW nor the US Justice Department or the environmental agency EPA were willing to comment on the report.
But a judge in San Francisco had given both sides until midnight today to reach an agreement, otherwise it would have gone to trial.
And the full details of the compensation and fines facing the German group had not yet been worked out.
"It's more a broad framework defining the key points which will be worked out and finalised in the coming months," the newspaper quoted an informed source as saying.
Bloomberg Business News reported that VW had agreed to set aside "at least USD 10 billion" to resolve civil claims by the US government and lawsuits by US car owners.
The crisis led to the departure of chief executive Martin Winterkorn and caused VW to delay releasing its 2015 earnings due to uncertainty over the costs of the scandal, which are expected to run into many billions of euros.
It has already set aside six billion euros in provisions, but the final sum is expected to be much higher.
The German giant, which has abandoned its ambitions of becoming the world's biggest carmaker ahead of Toyota in the wake of the scandal, has already started recalling some eight million vehicles affected in Europe.
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