Real had agreed a deal to sign De Gea on Monday, with Keylor Navas due to move in the opposite direction, but the paperwork was not submitted in time for the transfers to be completed.
It had been suggested that Real would appeal to FIFA in an effort to push the deal through, but there appeared to be little prospect of success and they indicated on Tuesday that the Spain international will remain at Old Trafford.
La Liga confirmed on Tuesday that De Gea was not among the players who transferred by Monday's deadline, but did not say whether Real Madrid actually attempted to register the player or if something went wrong to keep the deal from happening.
However, the statement implied that United were responsible for failure to beat the midnight deadline in La Liga.
The statement read: "After what happened yesterday in relation to the intent to sign the player David De Gea, Real Madrid wants to specify:
"1. Manchester United didn't open any negotiations for federation rights over David De Gea until yesterday morning.
"2. Real Madrid, despite the difficulties regarding entering into an operation of this kind on the last day of the transfer window, Real Madrid agreed to start these conversations.
"3. When Manchester United opened this negotiation yesterday morning they did so with the plan to reach an agreement with Real Madrid player Keylor Navas so that he would join the British club this season and said that they are in contact with the player's representative.
"4. Real Madrid and Manchester United reached a quick agreement over the transfers of both players. After drafting the corresponding contractual documents with the goal of proceeding within the time frame of both the TMS of FIFA as well as La Liga, Real Madrid sent the contracts to Manchester United at 13:39 Spanish time.
"5. Manchester United replies with its comments eight hours later at 21:43 Spanish time, including small modifications. Because they were not major, all were immediately accepted by Real Madrid with the intention to register the player in both TMS and the Liga in time.
"6. Real Madrid, after getting the signatures of both De Gea and Keylor Navas, remitted the papers to the English club with the contracts signed at 23:32 Spanish time, and waited to receive the signed documents from Manchester United.
"7. Manchester United reached a final agreement with Keylor Navas' representative at 23:53 Spain time and it was at that time that the documents were remitted to player to be signed.
"8. Manchester United put the data regarding the David De Gea transfer into the TMS but not that of Keylor Navas at 00:00 Spanish time, while remitting to Real Madrid the signed transfer contracts at the same time. Real Madrid received that complete documentation at 00:02 and tried to access TMS but it was already shut down.
"9. At 00:26 Spain time, FIFA's TMS system invited Real Madrid to add the information about the player David De Gea due to the fact that England's transfer window remained open until today. Real Madrid, realising the eventuality of a contentious run for the player's transfer, decides to remit the contracts to La Liga, even knowing that the window for the deal had closed.
"10. Definitely, Real Madrid did everything that was necessary at all times to get these two transfers completed."
Sources earlier told ESPN FC that United are happy to keep De Gea for the final year of his contract and believe he could effect a return to form and favour in the way Cristiano Ronaldo and Wayne Rooney have done after transfer sagas in the past.
De Gea has not played for United since July due to concerns that he would be distracted by the uncertainty over his future, with summer signing Sergio Romero playing every game so far and 22-year-old academy graduate Sam Johnstone serving as backup.
Louis van Gaal has yet to include either De Gea or Victor Valdes in a match-day squad so far this season, with the latter also seemingly set to remain at Old Trafford after his transfer to Besiktas fell through.
Anders Lindegaard, though, has joined West Bromwich Albion on a free transfer.