Several messages between Mr Giggs, 48, and Ms Greville, 38, which are believed to have been shared during their initial days of romance, were read out by his attorneys at the Manchester Crown Court on Thursday.
On 30 August 2017, the public relations executive had messaged Mr Giggs that he made her "heart flutter because you are a nutter".
She added: "Every day you do me proud not just because you are well endowed."
The next day, Mr Giggs responded with an innuendo-laced poem that began with "Darling Kate".
It read: “Unequivocally our love was fate. I fell in love with you at first sight. I remember coz I was as high as a kite.
"Those beautiful eyes, made me shiver.”
"I'm not gonna lie, I think of you, I dream of you. Can't help thinking, pulling you was my greatest ever coo.
"That stomach, those abs, those pictures you sent.
"So I can keep tabs on you, makes me feel funny down there. Especially when you are there and you look up and stare.
"I'm beginning to think you are always right. That's okay it will keep us tight
"I'm gonna end by saying you are my love, my friend, my soul.
"And most of all you believe in me. Which makes me feel as hard as a totem pole."
The footballer is on trial for allegedly headbutting Ms Greville at his home in Worsley on 1 November 2020, along with controlling and coercing her during their relationship from August 2017 to November 2020.
During the trial on Tuesday, the self-confessed “love cheat” footballer told the jury that he had been unfaithful in all his previous relationships, but had never assaulted a woman.
The court was also told that Mr Giggs said he loved his former girlfriend "more than all my Premier League appearances, which is a lot".
On Christmas Day 2016, Mr Giggs wrote: "Babe, make the most of Christmas because the next 40 years will be spent being pampered.
"Every day will be like Christmas waking up together with you. You are my angel. I'm the luckiest man alive to have ever met you."
Ms Greville replied: "I can't wait to spend the rest of my life with you. I love all of you. Your beautiful mind. The fire in your soul."
On New Year's Eve in 2016, Mr Giggs wrote: "You have got me through the most difficult period in my life. Thank you for putting up with me because I'm not perfect. Far from it. You are my babycakes."
In April 2017 he wrote: "Morning, I have to continually ask myself this question. How the f*** did I pull you? You are a painting, a thoroughbred, a supermodel, an absolute dream."
Ms Greville replied: "Ha ha. You're my world Giggsy baby."
Additional reporting from PA.