Simon Danczuk, who represents Rochdale in north west England, allegedly exchanged explicit text messages with the 17-year-old girl, the 'Sun' reported.
The pair contacted each other over a period of months after the girl, now 18, contacted the 49-year-old lawmaker about a job in his constituency office.
Danczuk then sent the student a number of flirty messages.
Detectives from Greater Manchester Police investigated the case to ensure that the student was not being "harassed".
"On 29 December, Greater Manchester Police received a report from a member of the public that a man from the Rochdale area had been communicating inappropriately with a teenage girl. The matter was looked into and it has been determined that no offences have been committed," a spokesperson said.
"Today's 'Sun' story, while not entirely accurate, refers to an extremely low point in my life. My behaviour was inappropriate and I apologise unreservedly to everyone I've let down. I was stupid and there is no fool like an old fool," he tweeted.
The MP was recently dumped by his girlfriend, Councillor Claire Hamilton, and last year he had a very public split from his wife, Karen Danczuk, known as the "selfie Queen" in the British media for her obsession with posting selfies on Twitter.