The scandal centres on Park's shadowy confidant, Choi Soon-Sil, who is accused of using her ties with the president to strong-arm local firms into donating millions of dollars to two non-profit foundations Choi then used for personal gain.
"We need to question the president Tuesday, or Wednesday at the latest," Yonhap news agency quoted an official with the Seoul prosecutors' office as saying, adding they had sent a notice to her office and were waiting for a response.
Park could respond as early as Tuesday after she appoints a lawyer to represent her, her spokesman said without elaborating further.
If Park agrees, she will be the first South Korean president to be questioned by prosecutors while in office. She had earlier vowed not to hide behind presidential privilege if required to give testimony.
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Park faces allegations that she pressured the heads of the country's powerful conglomerates to donate money to Choi's foundations during a meeting with them last July.
Samsung scion Lee Jae-Yong and Hyundai Motor chairman Chung Mong-Koo were reportedly among those who attended that meeting.
Prosecutors quizzed Chung over the weekend and were currently questioning Lee, Yonhap said.
The "donations" from the firms amounted to nearly $70 million, including 20 billion won ($17.5 million) from Samsung and 12.8 billion won from Hyundai -- the South's top carmaker.
Samsung -- the world's top maker of smartphones -- is also accused of separately offering millions of euros to Choi to bankroll her daughter's equestrian training in Germany.
Choi, whose father was an elusive religious figure and a longtime mentor to Park until his death in 1994, was arrested earlier this month for abuse of power and fraud.
The 60-year-old is also accused of meddling in state affairs to the extent of nominating officials and editing Park's speeches even though she has no official title or security clearance.
Nationwide fury
The scandal that emerged late last month has sparked nationwide fury with tens of thousands taking to the streets to call for Park's resignation.
The latest rally on Saturday drew one million people, according to organisers, making it the largest public protest in South Korea in nearly three decades.
The protest -- held only a few blocks away from her office -- made Park "deeply aware of the gravity of the situation", her spokesman said.
"President Park heard people's voices with a heavy heart yesterday... she is thinking hard on ways to fulfil her responsibility as president and to normalise state affairs," Jung Youn-Kuk told reporters.
Two of Park's former aides have also been arrested over abuse of power or fraud after being accused of collaborating with Choi to extract money from companies or leaking sensitive state documents to her.
The scandal has seen Park's approval ratings plummet to five percent -- a record low for any serving South Korean president -- with many mocking her for being Choi's "puppet".
