"He makes a fleeting reference to jamming with the then-fledgling Eagles in Los Angeles in 1971. But he downplays just how eager the soon-to-be high flying American band was to have him join their ranks, an experience he described in greater detail in a 1996 Union-Tribune.
“I was in Ian Matthews’ band, and I think we met when they weren’t quite the Eagles — they were still Linda Ronstadt’s band — at Duke’s Diner,” Thompson recalled in that interview.
“It was something like: ‘Hey! You English guys must come to our rehearsal place, and you’ll play for us and we’ll play for you.’ Ian and I had a listen, and the Eagles sounded pretty stunningly tight, playing things from what would be their first album. And that was all fun. Then we ‘hung out’ a bit, as they say, on tour. I only heard about it (being considered as an Eagle) later from management people. I never was directly approached.”
However, during a a 2017 Union-Tribune interview, Fairport Convention alum Matthews said: “The Eagles didn’t really give me a second look, but they were clearly very, very interested in Richard.”
https://www.sandiegouniontribune.com/entertainment/music/story/2021-04-18/richard-thompson-a-quiet-legend-pens-revelatory-memoir-it-was-necessary-to-write-about-the-hooker
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I wonder if he would have joined The Band, if asked? He was certainly a big fan and their influence on leading Fairport Convention to develop British folk-rock is undeniable. But just look at the joy on his face here as he plays with Garth Hudson...