Former premier Yousuf Raza Gilani has said he would prefer going to jail rather than appear before Pakistan's anti-corruption watchdog that is investigating graft charges against him.
Meanwhile, Maryam has rejected an offer for better facilities in jail.
Three cases -- Avenfield properties case, Flagship Investment case and Al-Azizia steel mills case -- were launched by the National Accountability Bureau on September 8, 2017 following a judgment by the apex court that disqualified Sharif.
Maryam, who is being groomed as Sharif's political successor, appeared in the court for the first time on Monday.
The two were convicted on July 6 in the Avenfield properties case linked to the Sharif family's ownership of four luxury flats in London.
The banned Jamaat-ud-Dawah head has hired Lahore-based law firm -- Mirza and Mirza Law Associates -- to file the petition.
Abbasi will be sworn-in in later at a special ceremony at the President's House.
The Panama verdict will decide how Pakistan handles its leadership problems.
Sharif is in London with his ailing wife Kulsoom, who is suffering from throat cancer and has undergone three surgeries so far.
Pakistan's former President Asif Ali Zardari was on Monday summoned by a court to appear before it on October 29 in relation to five corruption cases against him which were re-opened last week, compounding his legal woes.
Authorities arrange helicopters to shift Sharif to jail; 300 PML-N workers arrested.
The Pakistani Supreme Court ordered for a joint investigation team to probe the allegations of money laundering against his family.