The football chief is also alleged to have texted about "boring" training and discussed booking a hotel for the pair while at a session, The Press Association reports.
The paper's sister publication, The Sun on Sunday, is currently subject to the injunction involving a married celebrity who wants to keep his and his spouse's names out of a story about his alleged extra-marital sex.
The Sun said the football boss at the centre of the latest story threatened them with an injunction after learning the relationship was set to be exposed.
The newspaper reported that he asked for the woman to send him pictures of herself in high heels, stockings and without underwear as a reward for his team winning an away game.
He also told her that the fact he was in the public eye was problematic as he was also in a partly loveless and sexless marriage.
One of his text messages allegedly said: "I'm churning out a boring footie session for my boys.
"Did cross my mind if sex is going to be a part of this first date scenario.
"A famous madame died yesterday and her advice to women was never have sex on the first date. Stupid cow. I'm pleased she's gone."
The celebrity at the heart of The Sun on Sunday's injunction, known only as PJS, is waiting to find out if he has won a battle at the Supreme Court.
On Thursday he appealed against the Court of Appeal's decision that the order barring The Sun on Sunday from naming him should be lifted.
The injunction was granted to protect the children PJS has with his world-famous spouse, identified only as YMA.
A panel of five justices headed by Lord Neuberger reserved their decision and said the injunction in the case would remain in place in the meantime.
They said they would deliver their judgment at "a later date" but did not specify when this was.
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