Yukos was once Russia's largest oil producer and regarded as one of the country's best-run companies.
It was sold off in auctions and ordered by the state to pay off billions of dollars in back tax claims.
Mikhail Khodorkovsky, the head of Yukos, was earlier charged with fraud and tax evasion and sentenced to eight years in prison.
However, his supporters say he was prosecuted because he funded opposition parties and the Kremlin wanted to stifle his political ambitions.
Khodorkovsky began a hunger strike on January 30 to protest the authorities' refusal to give proper Aids medication to Aleksanian.
Khodorkovsky accused officials of trying to extract incriminating and false confessions from Aleksanian and denying him treatment until he co-operates.
The prosecution raised no objections to suspending Aleksanian's trial, but insisted he must remain in custody.
Aleksanian is charged with embezzling 12 billion rubles ($490 million) of funds and shares in the Yukos subsidiary Tomskneft.
On Tuesday Aleksanian renewed accusations that prosecutors are seeking to pressure him into testifying against other former Yukos executives.