The incident, widely reported in the local media, comes at a crucial time for Islamists rallying behind Muslim Brotherhood candidate Mohamed Morsi, who is set to face former regime figure and ex-military man Ahmed Shafiq in a June 16-17 vote.
The allegations are especially controversial for ultraconservative Salafi Muslims, who advocate a strict interpretation of Islam and the segregation of unrelated men and women. The accused lawmaker and sheikh, Ali Wanees, ran for parliament on a coalition led by the Salafi-led Nour Party, which controls 25 percent of parliament.
Earlier in the year, the party was forced to issue an embarrassing and apologetic statement after one of their lawmakers was reported to have lied to cover up a nose job, claiming that he sustained injuries to his heavily bandaged face from a carjacking and beating. He was kicked out of the party and resigned from parliament.
This time, though, Salafi leaders are backing Wanees, calling the police report "fraudulent." He could not be immediately reached for comment.
According to security officials with access to the police report, four police officers say a vehicle was parked in a dark area along an agricultural road linking the city of Banha to Cairo on Thursday evening.
In their report, the officers say they approached the car and saw a woman in a full face veil sitting on top of a man who was touching and caressing her. They say that when they knocked on the car window and demanded the man show his national ID and car registration, he yelled back, "I am Ali Wanees, member of parliament, you sons of dogs."
Wanees was briefly detained before police realized he was a lawmaker and has parliamentary immunity. The officers filed a report against Wanees, saying that he was caught "violating public decency." Egypt's MENA news agency say state prosecutors have requested that his immunity be lifted so they can investigate.
In a video posted on his website, Wanees says the 23-year old woman is his niece and that the two had been running errands when she fell ill, forcing him to pull the car over.
"I was surprised to find someone knocking on the car window," he said. "I opened the door and was surprised to be asked for my ID and I unfortunately had really forgotten my ID, the car registration and my parliamentary ID."
He says there was a dispute with the police at the scene and that after around 30 minutes he was allowed to leave.
Salafi religious leader Sheik Farid Abdel-Ghofar was quoted on Wanees' website saying that Shafiq's presidential campaign was to blame for the distortion of facts.
"Shafiq's campaign is the one that published these words to defame the image of Islamists," he said, adding that the police report was "fraudulent."
Ex-air force commander Shafiq is widely regarded as the candidate of choice for the country's military rulers and its senior police. His campaign could not immediately be reached for comment.
Adding to the political tension is uncertainty about the incoming president's powers, which are yet to be defined. Egypt's old constitution was suspended after the overthrow of Hosni Mubarak during the 2011 uprising.
The writing of a new constitution was delayed after liberals walked out of a panel tasked with its drafting earlier this year. The Islamist-majority parliament had initially selected a panel dominated by Muslim Brotherhood supporters and Nour Party lawmakers.
On Saturday, in what appeared to be an attempt to allay concerns among liberals, Brotherhood member and head of parliament Saad El-Katatni vowed that no political group will be able to force its opinion on the 100-member panel that will be selected Tuesday.
"The panel will write a constitution that speaks for all Egyptians," he told reporters.
Earlier in the week, the country's ruling military generals and 22 political parties agreed that Islamists will only take half of the new panel's seats and that the rest would go to legal experts and representatives of unions, government ministries and the Coptic Church.