The 64-year-old Mueller has two years left on his current contract.
Shares in Volkswagen jumped on the news, and closed up 4.5 percent at 171.58 euros.
Under Mueller - who was promoted to the top job in the chaotic days following the public disclosure of the diesel cheating - VW has weathered the blows from the scandal while at the same time embarking on an aggressive expansion into electric cars. The scandal has cost the company over $30 billion. Amid opposition from labour leaders, Mueller failed to sell motorbike maker Ducati past year.
The frontrunner for the CEO job, Mr Diess, joined VW from BMW in 2015.
Volkswagen said in a short statement that board of directors Chairman Hans Dieter Poetsch was in discussions with top managers about their duties and that the result was "currently open". Muller's dismissal comes despite the company's strong performance last year which saw profits of €17bn (£14.8), a 16 per cent increase on the year before.
The issue with restructuring VW lies with a tug a war between interested parties including the controlling families, stakeholders, and unions.
Nestle wants to make all of its packaging recyclable or reusable by 2025, the Swiss food giant said yesterday, becoming the latest food company to vow to cut plastic waste. The company also said that Muller "showed his general willingness to contribute to the changes".
Matthias Müller is likely to be replaced as chief executive of Volkswagen as the carmaker struggles to move on from the diesel emissions scandal that claimed the scalp of his predecessor.
Kilian, who works directly under labor boss Bernd Osterloh, will replace personnel chief Karlheinz Blessing, the sources said, giving workers representatives a direct say on strategy and cost-cutting.
"If Diess is confirmed as the successor, VW shares will extend their gains", Evercore ISI analyst Arndt Ellinghorst told the publication, who has an "outperform" rating on the stock. According to the company, final decision is expected to be made by the end of the week. German newspaper Der Spiegel has reported that the changes could take place as soon as Friday.
257 dead in Algerian military plane crash
Algeria's defence ministry was quoted as saying that 247 passengers and 10 crew members were among the dead following the crash . The Ilyushin IL-76 transport plane was bound for Tindouf in southwest Algeria near the borders with Morocco and Western Sahara.