The deadline has been extended to May 2 from April 18.
The case is widely viewed as a political wildcard.
Ms Yingluck and her cabinet could be removed from even caretaker status, leaving open an opportunity for a neutral prime minister to step in.
Both pro- and anti-government protesters are closely watching the developments in this case. The anti-government group has made no secret of their wish to seize ruling power in the event Ms Yingluck is deposed, while pro-government goups have vowed to prevent it.
The case was filed by a group of senators led by appointed senator Paibul Nititawan. They asked the court to decide the prime ministerial status of Ms Yingluck after the Supreme Administrative Court ruled her decision to transfer National Security Council chief Thawil Pliensri in 2011 had been unlawful and unfair because she had not given appropriate reasons for Mr Thawil's transfer.
Ms Yingluck argues her prime ministerial status ended when she dissolved the House of Representatives on Dec 9, 2013 and that she remains in office only in a caretaker capacity because Section 181 of the 2007 Constitution requires her to do so until a new cabinet is in place.