Henning Berg, the Legia manager, declared on Thursday that the Swiss-based court could decide on their bid to overturn their effective expulsion from the competition by the end of the weekend.
But his chairman then claimed UEFA had failed to co-operate with their attempts to force a fast-track hearing and CAS had received nothing from the Polish club just before the close of business on Friday.
UEFA's appeals body on Thursday upheld a decision by its disciplinary body to award Celtic a 3-0 third qualifying round second-leg win after the Poles fielded an ineligible player.
That allowed Celtic to step up preparations to face Maribor in Slovenia on Wednesday in the play-offs, while Legia look set for a Europa League clash at Aktobe in Kazakhstan on Thursday.
In a statement just before their offices closed, CAS said: "If an appeal is filed today before close of business, it seems unlikely that a hearing can still be organised on Monday.
"The only option for Legia would be to request urgent provisional measures. UEFA would then have a short time limit to send a written answer and the CAS would issue an order granting or rejecting the request, probably on the same day (Monday).
"If the request is rejected, there will be no change in the programme of the UCL (UEFA Champions League) next week.
"If it is granted or partially granted, we should see what is ordered and it is too early to make any hypothesis."
Legia chairman Dariusz Mioduski earlier said that UEFA would not fast-track their application.
He told BBC Scotland: "A few days ago we requested UEFA for a quick appeal and we also discussed the possibility of not getting the right result.
"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 and the governing body made no further comment.
The punishment, which turned a 6-1 aggregate win for the Polish side into an away-goals defeat, stemmed from Legia's failure to register Bartosz Bereszynski in their squad list for the second qualifying round against St Patrick's.
That 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.
Legia player Henrik Ojamaa declared the punishment excessive as he was unveiled at Motherwell following his loan move.
Ojamaa, who was on the bench for both ties, said: "At the end of the day everybody understands that rules are rules but everybody feels the punishment was very excessive.
"It's a UEFA decision and the responsibility lies with them. The club realise that they made a mistake but the point was the punishment shouldn't be as severe because the team have done very well to get through convincingly and to get that taken away from them is obviously a big disappointment."