A challenge by a third man, whose case was rejected at the High Court on the same day as Mr Nicklinson's, will also be heard today.
The locked-in syndrome sufferer, who cannot be identified for legal reasons but is referred to as "Martin", suffered a massive stroke in August 2008.
He is unable to speak and virtually unable to move, describing his life as "undignified, distressing and intolerable", and wants to be allowed a "dignified suicide".
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The former Labour minister criticised the government's approach to leaving the European Union.