The Blues striker, who helped fire Antonio Conte’s side to Premier League glory last season, has fallen out with the Italian.
Costa was linked with a move back to his former La Liga club, but Atletico were slapped with a transfer ban, reports the Liverpool Echo.
The Spaniard has been forced to stay on Chelsea’s books, despite Conte signing Alvaro Morata this summer.
Thompson thinks Liverpool are well placed to solve the conundrum – by loaning the Spain international for half of the season!
He told the Irish Examiner: “Dare I say it: Diego Costa signs for Atletico Madrid and then gets loaned out to Liverpool?
“It would be quite a dream, wouldn’t it? I’d take him for six months.”
Despite being heavily linked with a move to the Chinese Super League back in January, Costa fired Chelsea to the Premier League title with 20 goals, and Thompson would love to see the combative striker at Anfield.
“Going back three or four years when Liverpool were in for Costa, I think we offered £20m,” he added.
“It was £25m and the deal never got done. I was amazed. If we’d have had Suarez and Diego Costa, God, they’d have frightened everybody.
“You love these players. Every other supporter moans at them but you know as a Chelsea fan how much Costa will fight for you, do the battling for you. Good skills too. Good technique.
“Is Morata going to be (as effective) when you get into battles in the Premier League?
“It didn’t matter whether you were playing against Hull City or Manchester United, you knew what you’d get off Costa.”