Legia manager Henning Berg claimed on Thursday night that CAS could overturn their effective Champions League expulsion in time for Celtic to be sent to play in the Europa League next week.
UEFA's appeals body had earlier upheld a decision by its disciplinary body to award Celtic a 3-0 second-leg third-qualifying-round win over Legia after the Poles fielded an ineligible player.
That allowed Celtic to step up preparations to face Maribor in Slovenia on Wednedsay in the play-offs, while Legia look set for a Europa League clash at Aktobe in Kazakhstan on Thursday.
But the Poles vowed to "promptly" take the case to the Swiss-based CAS, which later confirmed it could hear the case quickly.
However, Legia chairman Dariusz Mioduski said on Friday morning that UEFA would not fast-track their application.
"A few days ago we requested UEFA for a quick appeal and we also discussed the possibility of not getting the right result," he said.
"We received indication and declaration that UEFA would support us in a fast-track proceeding in front of CAS in case there was a rejection in our appeal.
"We tried to contact UEFA because we've prepared the application for fast-tracking but after several hours we finally got an indication that UEFA will not support us.
"This effectively means that they want us to accept their decision. We are very surprised by this change of position."
A UEFA spokesman said that they had not been given any information on the case from CAS. The Swiss-based court itself was unable to provide any update on Legia's case and whether they had submitted their appeal.
Former Rangers player Berg had earlier said: "We still have good hope of playing in the Champions League. CAS has made many different decisions to UEFA.
"We don't know who we will play if CAS make a different decision. Before the draw we were seeded, so the fair thing for us it to play Steaua. Celtic should play Aktobe and replace us, and Maribor should play Ludogrets. If we have all booked a charter, we can just change the destination.
"But the most important thing is to get CAS to make a decision. Normally they will do this within a very, very short time because this is an urgent matter.
"Also, in terms of legal approaches, if we are denied the possibility to seek an independent instance for this to be decided through time, then it will not be justice.
"We really really hope and we think that CAS will be able to do this over the weekend and have a decision in good time before these games are played."
Berg revealed Legia only listed 23 players of the 25 allowed in their squad for the second round against St Patrick's - the omission of Bartosz Bereszynski meant his three-match suspension had not officially been served when he came on as an 86th-minute substitute as Legia beat Celtic 2-0 at Murrayfield following a 4-1 first-leg success.
Berg said: "If you think about how many mistakes football clubs have made in the last 20 years, big and small, I can't remember any club who has been harder punished for a small mistake. A club that acted in good faith, that did not hide their cards.
"If people still think this decision is the right decision then it's a tragedy for football. Paragraphs and money will be the only thing that's important. I don't want football like that and I know most people don't want football like that.
"I've seen West Ham playing with (Carlos) Tevez for many games and he scored so many goals and they stayed up. He shouldn't have played because he was owned by a third party. Were they relegated? No, they were given a fine.
"Sunderland last year used players that were ineligible to play. Did they lost those games 3-0? No, and they stayed in the Premier League by the skin of their teeth."
Sunderland played striker Ji Dong-won in five games despite the fact they had not received international clearance from the German FA following a loan spell, including a cup tie on their way to the Capital One Cup final.