Lee, vice chairman of Samsung Electronics Co., is accused of giving or promising to give some 43 billion won ($36 million) worth of bribes to Park's jailed friend Choi Soon-sil in return for the state-run pension fund's backing of a merger of two Samsung affiliates, the team's spokesman Lee Kyu-chul said.
According to the prosecutors, Samsung signed a 22 billion won consulting contract in August 2015 with a Germany-based firm owned by Choi and allegedly sent the company billions of won, which was used to fund Choi's daughter's equestrian training, Yonhap News Agency reported.
The money that was originally promised to be handed over was included in the amount deemed as bribes.
Some 20.4 billion won the group donated to two non-profit foundations, allegedly linked to Choi. It was the largest amount given by any business group to the organisations.
Prosecutors suspect Samsung supported Choi in return for the National Pension Service (NPS) approving the contested merger of two Samsung subsidiaries on July 17, 2015.
The Samsung scion has denied most of the allegations raised during a parliamentary hearing held last month. Investigators have accused him of perjury.
"In seeking the warrant, the investigation team concluded that establishing justice was more important than the possible impact it (the arrest) could have on the national economy," the spokesman said.
The decision came after Lee was grilled by investigators for about 22 hours last week.
The Seoul Central District Court will hold a hearing on Wednesday to review the legality of detaining the Samsung business chief.
The probe team has also decided to carry out the investigation on three other senior Samsung executives, including vice chairman Choi Gee-sung, without detention.
Also on Monday, the Constitutional Court held its fifth hearing of Park's impeachment trial where Choi appeared and denied any wrongdoing in her interactions with the President, Yonhap News Agency reported.
The Constitutional Court, the investigation team and a district court have all accelerated their proceedings to draw a swift resolution to the case. Millions of people have staged candlelight vigils for the past 12 weekends, demanding Park's prompt ouster.
The top court has until early June to decide whether to uphold or reject the impeachment motion, which Parliament passed on December 9.