Erdoğan, 58, is already the most powerful politician in Turkey, thanks to a blend of economic liberalism and religious, social conservatism that has landed his Justice and Development Party (AK Party) three consecutive election victories since 2002.
He favours switching to a presidential form of government similar to France and it is an open secret that he covets the post himself, since under party rules he cannot seek another term as premier when the next general election falls due in 2015.
At present, Turkey's parliamentary system vests more power in the prime minister while the president is largely a figurehead role, although choosing Constitutional Court judges is among his prerogatives.
Any changes would be written into a new constitution that the government began drafting this year to replace one written after a military coup three decades ago.
Erdoğan told the television channel ATV in an interview late on Wednesday that the president should belong to a political party. Currently, presidents must be non-partisan, and Gül had to quit the AK Party to take the post.
"We (will) have a president elected by the people, not one chosen by parliament. A president who is elected in such a strong way will transform this apparent two-headed system into a one-headed system," Erdoğan said, describing previous Turkish presidents as "window dressing".
Critics say Erdoğan shows autocratic tendencies, and fear they would become stronger with presidential powers.
In next week's hearing, the Constitutional Court will review a challenge by the main opposition Republican People's Party (CHP) to a law passed in January that extends Gül's tenure to 2014 and bars him from seeking re-election, according to a statement on the court's website.
If the court upholds the law and Gül's term expires in 2014, it would allow Erdoğan to step down as prime minister a year before his time is up, and seek election as president.
Bringing the presidential election forward would affect Erdoğan's political schedule, which has appeared geared to moving from the prime minister's office to the presidency in 2014. It would also potentially force a vote for a new president before the government has had time to propose constitutional changes granting far-reaching powers to the president.
Analysts at Eurasia Group said Erdoğan could easily win a presidential election, but his switchover could create political instability in Turkey.
"Erdoğan's shift into the presidency at an earlier date than planned could leave an unexpected power vacuum at the top of the AK Party, and Erdogan's successor may have difficulty keeping the party's various factions in alignment," the Eurasia analysts wrote in a note.
Gül served as Erdoğan's foreign minister before being elected as president in 2007 by a parliament dominated by AK Party lawmakers. He had held the premiership for four months before stepping aside in 2003 to make way for Erdoğan.
The controversy over the length of Gül's term goes back to his election for a seven-year term in August 2007.
Less than two months later, Turks voted in a referendum brought by the AK Party to change the law on presidential elections to make it by popular vote for five years, with the possibility of re-election.
Since then, there has been debate on whether the length of Gül's presidency is guided by the old law or the new. In January, parliament passed a bill that aimed to resolve that debate by including a temporary clause that says Gül's tenure is seven years and that he cannot seek re-election.
The CHP applied to the Constitutional Court to strike down that clause, alleging that it violates equality before the law by singling Gül out.
A Constitutional Court rapporteur has recently submitted a report on his opinion regarding the CHP's appeal, in which he said Gül's tenure is seven years and that he could be re-elected.