Grainger, who won in the double sculls with Anna Watkins at the London Games last week, had been told by five-time Olympic champion Sir Steve Redgrave that his friend Beckham would take her out if she won gold.
The English footballer came through on that promise but Grainger has been left with a voicemail rather than a dinner date.
She said: "It started off as a joke. Steve was talking about the lighting of the Olympic cauldron and the fact he was going to collect the torch from Beckham.
"I said, 'you've got to get me a meeting with him'."
Redgrave told the rower that Beckham had agreed on the golden provision of success.
Grainger said: "After I won it was all still just good fun – then I got a message on my phone from David Beckham.
"I'm kicking myself because I didn't take the call. It was an un-known number and I was at a hockey match and sitting in front of a four-piece band – I couldn't have spoken to anyone anyway."
Grainger insisted a meeting with Beckham "would be all good fun because he is a happily married man".
She added: "It has ended well because I have the message on voicemail and I let my mum hear it. At the end he gave his love to my family. So spread the word, my family are friends with David Beckham."
She said the former England football captain, who missed out on selection for the Team GB squad, said it would have been "an honour" to meet Grainger.
She added: "He could not have been nicer. I didn't get a number, so now I just guess I'll wait for the elusive phone call to come through from a mystery number."
After inadvertently snubbing the football legend, Grainger didn't miss the chance to meet perhaps the greatest player ever.
She was on hand to see Pele presented with an honorary degree from Edinburgh University yesterday in recognition of his contrib-ution to humanitarian and environmental causes as well as his football achievements.
Edinburgh Uni former student Grainger met the Brazilian hero at a reception at the Victoria & Albert Museum in London.
The 36-year-old Glasgow-born rower also talked about her meeting with Prince Harry when both watched the triumph of Sir Chris Hoy at the velodrome this week.
Grainger, who won gold after three successive Olympic silvers, said of the prince: "He was very friendly. He is passionate about sport and loves the fact he can be here, being part of it.
"He talked about the cycling but he also talked about my experience and said I must be grateful now I have got the gold."
She added: "The nice thing is that the big names who come and watch sport are not there to be seen.
"They are genuinely there to watch the sport and be part of it.
"Everyone gets that. I have a friend who was sat behind Helen Mirren in the athletics and the actress was gushing about how wonderful women's sport was doing.
"She was someone who really knows what was going on and was interested."
However, Grainger said the most memorable meetings of the week had been with members of the public at the Olympic Park.
She said: "That has thrown me far more than anything else. I have never been recognised before, I have never been stopped by anyone who has known who I was.
"It is happening a lot now and everyone who does is just so warm and wanting to give me hugs.
"That has been really humbling."