It was the Australia who came out on top in their only previous encounter, six years ago at Queen’s, but Murray continued his strong start to the grass-court season to win 7-6 6-3 and reach his final final on home soil since 2016.
The two-time Wimbledon champion flew out of the blocks in incredibly hot conditions, breaking early and taking a swift 3-0 lead in the opening set. Thompson got it back on serve though and took it a tie-break, which Murray sealed with an ace.
The second set was a more routine affair, Murray moving 5-1 up and giving himself the breathing room that meant a break from Thompson only slighty delayed the end of the match, with the British number three getting the job done on his third match points.
Murray will face either Jurij Rodionov or Zizou Bergs in Sunday’s final, before turning his attention to Nottingham next week after taking a wildcard to play in the Challenger event.
“Nice to get through in straight sets today,” Murray said.
“It was a very, very tight first set. I improved a bit in the second set, served well and starting the hitting the ball a bit better from the back of the court.”
On his overall form, he added: “The last couple of matches have been good. Jordan and Jason Kubler are very good grass-court players. Very experienced on the surface.
“To come through both of them is very positive and I get the opportunity to play in a final tomorrow, which will be great. It’s been a while since I’ve won a tournament on home soil. Hopefully I can do that tomorrow.”