The former footballer was sacked from his presenting roles on The One Show and Match Of The Day after junior staffers raised concerns over his behaviour.
And Jenas now fears for the future of his marriage with his wife of 13 years, Ellie Penfold, who is said to be angry but has not ended their relationship.
The 41-year-old ex-England, Newcastle United and Tottenham Hotspur midfielder said: ‘I haven’t slept properly since they told me, to be honest.
‘I mean, the anxiety that builds up in your body over these types of things. I haven’t had any sleeping pills. I’ve just not been sleeping.
‘You just kind of lay there, mind’s ticking over until you just pass out. Part of me would just rather go out and run until I collapse.’
Jenas, who has three children with Ellie and another teenage daughter from an early relationship, revealed his wife has not yet told him whether their marriage can last.
He also told the newspaper: ‘When I see Ellie most days, she just kind of gets up. She goes to make sure the children are OK. I feel ashamed. I feel I’ve let everybody down – friends, family, colleagues.
Jenas previously told his former female colleagues he is ‘genuinely sorry’ for sending inappropriate messages – issuing a direct apology after being allegedly told not to contact the two women due to ‘legal process’.
The presenter previously said he sent ‘inappropriate messages’ but insisted they were ‘between two consenting adults’ and that his behaviour was not illegal.
In addition, he has criticised how his former employer handled his sacking and has said he will be speaking to his lawyers about the situation.
According to official figures published last year, Jenas earned £190,000 to £194,999 for his work on the FA Cup, Match Of The Day and the World Cup at the BBC.
His salary for working on The One Show was not in the public domain, as it is produced by BBC Studios.
It is the latest blow to the BBC after the Huw Edwards child abuse scandal and bullying allegations on Strictly Come Dancing.
Moments after it was announced he had been removed from all BBC presenting line-ups, Jenas began co-presenting a talkSPORT drive-time show with fellow former footballer Jermaine Pennant.