The Spain international continues to be heavily linked with a return to the La Liga outfit after being deemed surplus to requirements at Stamford Bridge.
Atletico are, however, working under the restrictions of a transfer ban which prevents them from registering new players until January.
That means that any deal for Costa could see a short-term loan switch sanctioned, with Thompson of the opinion that Jurgen Klopp should be asking questions if such a situation arises.
The former Reds defender 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.
"Going back three or four years when Liverpool were in for Costa, I think we offered £20m.
"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 [Alvaro] Morata going to be [as effective] when you get into battles in the Premier League?
Costa netted 20 times for Chelsea in the Premier League last season as they swept to title glory.
His overall record for the Blues has seen him net 59 times in 120 appearances, but Antonio Conte claims he took the decision to move him on in January and has spent £70 million this summer bringing in Morata from Real Madrid as his replacement.