FRANKFURT, Nov 27 (Reuters) - Volkswagen Chief Executive Herbert Diess has asked the families who control a majority voting stake in the world's largest carmaker to back a contract extension for him in a bid to break a deadlock, two people familiar with the matter said.

The appeal for support from the Piech and Porsche families comes after Diess was forced to relinquish responsibility for the VW brand in June to remain as group CEO.

"He is bringing the issue to a head," one of the sources told Reuters on Friday.

Another person familiar with the matter said Diess may review his position as CEO if his continued efforts to reform the multi-brand car and truck maker are stifled.

Volkswagen Group, which also owns the Bentley, Bugatti, Porsche, Seat, Skoda and Audi brands, declined to comment, as did the families, the works council and the German state of Lower Saxony, another stakeholder.

Diess was appointed head of the VW brand in 2015 and group chief executive in April 2018. His contract is due to expire in 2023.

German companies tend to deliberate over contract extensions only a year before they expire, but the appeal comes after Diess struggled to win backing for key reforms.

These included installing two allies Arno Antlitz as chief financial officer and Thomas Schmall as chief procurement officer on the management board, three sources told Reuters.

Diess has met opposition to these appointments from Volkswagen's powerful labour bosses who control half of the 20 seats on the company's supervisory board, or board of directors.

If labour leaders team up with Lower Saxony, which controls a 20% stake, they can veto significant decisions.

Rather than approving each individual appointment, the labour leaders insist on approving a "package solution" which is "harmonious", two people familiar with the deliberations said.

Volkswagen's labour chief Bernd Osterloh is also said to oppose a contract extension, one of the three sources said.

In a post on Linkedin this week, Osterloh said there was no fight about management appointments because no committee on the supervisory board had been formally consulted about the issue.

The issue would be decided "when the time is right" he said.

Diess, for his part, voiced his frustration in a column in the German daily Handelsblatt published on Friday.

"When I took office in Wolfsburg, I had firmly resolved to change the VW system. This meant breaking up old, encrusted structures and making the company more agile and modern," he said.

"Together with many companions with the same level of motivation, I succeeded in doing this in many places, but not in some, especially not yet at our corporate headquarters in Wolfsburg." (Reporting by Jan Schwartz and Edward Taylor Editing by Riham Alkousaa, Alexander Smith and Susan Fenton)