Volkswagen has reached a deal with authorities in the United States to settle the case over its cheating of diesel emissions tests that would involve it paying each affected customer $5,000 (€4,422), Germany's Die Welt newspaper reported.

Citing unidentified sources close to the negotiations, Die Welt said the agreement would be presented to Judge Charles Breyer in San Francisco tomorrow, avoiding a trial.

Volkswagen declined to comment.

Earlier today, the possibility of a settlement had boosted the carmaker's shares by 6.6%, the biggest gain in Germany's benchmark DAX index.

A US federal judge last month gave VW and regulators until tomorrow to agree on a fix for nearly 600,000 diesel cars on US roads implicated by VW's emissions test-rigging scandal.

The company does "not believe any expedited hearing or bench trial is appropriate or required", according to the agenda for the hearing at the San Francisco district court about VW's progress towards reaching a deal with the US Environmental Protection Agency.

The plaintiffs - a committee representing thousands of consumers who say they were tricked into buying polluting diesel vehicles - proposed an expedited hearing or bench trial, or an expedited "all issues" trial including punitive damages.

Die Welt said the agreement did not include a detailed plan to fix the affected cars, nor were details fixed on fines and other compensation measures, with one source saying the deal would be fleshed out in the coming months.

However, the owners of affected cars should receive $5,000each in compensation and VW will separately have to pay to fix their vehicles, the paper said.

German lawyer Christopher Rother told Die Welt that plaintiffs in Europe would seek to emulate the US deal for their customers too.

Earlier two sources familiar with the matter said VW would substantially increase the amount of money set aside to cover its emissions test cheating scandal from the €6.7 billion currently earmarked.

Europe's biggest carmaker will make provisions for a double-digit billion-euro amount in its 2015 results on 28 April, the sources said, speaking on condition of anonymity due to the sensitivity of the matter.

One said the German company might not pay a dividend to shareholders on the results and if it did, it would be less than €1 per share.

Volkswagen paid out €4.80 per common share and €4.86 per preference share on 2014 results.

The company is struggling to put a cost on the biggest business scandal in its history seven months after it admitted to cheating US diesel emissions tests, as it is embroiled in legal proceedings around the world.

Final provisions will depend on the outcome of talks with US regulators, the Justice Department and the Federal Trade Commission, one of the sources said.

Analysts have said a deal with US authorities, also on financial penalties, would remove a major deterrent to investing in VW, whose shares have lost billions in value since the scandal broke in September.