The 57-year-old actress said she received a surprise phone call from Mr Trump while she was filming Primary Colours in the early Nineties.
Thomspon said the businessman offered to host her at Trump Towers and treat her to dinner.
Speaking to Swedish talk-show host Fredrik Skavlan she said: “He asked me out once, Donald Trump. I was doing a film called Primary Colours with John Travolta in Hollywood.
“I was in my trailer and it was on the Universal lot and I’d just been divorced and my decree absolut had just come through the day before. So the phone rings in my trailer, which it’s never done before, and I look at it like a moose has entered my trailer.
“So I lift up the phone and hear: ‘Hi it’s Donald Trump here’. I said: ‘really? Can I help you?’
“And he said: ‘I wonder if I can offer you some accommodation in one of my Trump Towers, they’re really comfortable.
“And I said: ‘Why are you offering me somewhere to stay in my trailer?’”
Impersonating his response, Thompson said: “’Well you know I think we’d get on really well and maybe we can have dinner some time?’
“I didn’t know what to do with myself. I was on my own and said: ‘Well I’ll get back to you, thank you so much for ringing up.’”
Thompson said she regrets not taking him up on the offer, with Skavlan joking that she could have been the First Lady.
“I’d never met him and I’ve never met him since,” she said. “I wish I had. Think of the stories!”