The High Court confirmed a decision was due at 10:15 local time (09:15 GMT) on Friday.
Dewani, 31, denies accusations by the South African authorities that he arranged the contract killing of his Swedish wife Anni, 28, while the couple were on their honeymoon in Cape Town in November 2010.
Dewani lodged an appeal to Britain’s high court after Theresa May, the country’s interior minister, signed an order for his extradition following a judge’s ruling in August that he should be sent to stand trial.
At December’s hearing, Dewani’s lawyers argued that their client’s health and life would be at risk if he were to be extradited and that he was a serious suicide risk.
In November 2010, Dewani and his bride were being driven through a dangerous part of Cape Town when their taxi was hijacked. Dewani was thrown out of the vehicle while his wife, a native of Mariestad in central Sweden, was driven off and later shot dead.
Driver Zola Tongo was jailed for 18 years after turning state witness and pleading guilty to his part in the killing. He claimed Dewani had ordered the shooting and paid him.