The clubs have agreed a fee of £500,000 but the deal is stumbling on the Berkovic's wage demands.
Peter Storrie, Portsmouth's chief exective, and Berkovic's agent Pini Zahavi have been locked in talks to conclude the deal.
Storrie said: "The two parties have moved closer."
He added: "If the obstacles remaining can be overcome then we remain hopeful that Eyal will be a Pompey player by the weekend."
The Israeli has not appeared for City since being injured by Lee Bowyer in a substitute appearance at Newcastle on 22 November.
Despite City's poor run, Keegan refuses to consider Berkovic for selection, making it clear he has no future in Manchester.