He told the Sun the texts were a “huge error of judgement” for which he was “deeply sorry”, adding that he was “not a sex pest”.
Jenas, who is married with four children, told the paper he is receiving therapy and accepted he “fell below [the] standards” of the BBC.
The 41-year-old continued: “I feel so ashamed. I’ve let everybody down – my colleagues, my friends and, most importantly, I’ve let my family down.”
The issue was brought to the BBC’s attention a few weeks ago. The corporation announced his departure on Thursday.
A BBC spokesperson said: “We can confirm Jermaine Jenas is no longer part of our presenting line-up.”
The BBC has not commented since Jenas spoke to the Sun.
Jenas said he was sacked during a video call on Monday while on a family holiday, five days after being told of the allegations against him by the BBC.
In a lengthy and at times tearful interview on camera, Jenas said “nothing physical ever happened” but said he considered what he did “cheating”.
He continued: “I was given an incredible opportunity and I know it’s on me that right now I feel I have lost everything.
“I feel like people are judging me and that I am the number one target right now in the country.”
He denied sending any explicit pictures or video, the Sun reported.
Jenas was presenting on talkSport Drive when the news of his BBC dismissal broke on Thursday and repeatedly refused to be drawn on why he had been sacked.
The ex-Premier League footballer also works for TNT Sports as a pundit and co-commentator, and fronts their Formula E racing coverage.
Jenas earned between £190,000 and £194,999 at the BBC in the last financial year for his work on coverage of football including the FA Cup, Match of the Day and the World Cup.
His salary for his work on The One Show has not been made public because it is paid by BBC Studios, the BBC’s commercial production company, which doesn’t reveal how much it pays presenters.