Sources told ESPN that Arsenal had three bids rejected for Aston Villa's Douglas Luiz on Thursday -- the last of which was worth £25 million -- as they tried to bolster their midfield options in the wake of setbacks for Elneny and Thomas Partey.
Elneny has seen at least two specialists to examine the extent of the problem, although there was better news on Partey, who has been struggling with a thigh injury.
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"With Mo, probably it is a little bit easier [to give a timeframe] and we are talking months," Arteta told a news conference on Friday.
"With Thomas, we are talking weeks or even shorter than that, I hope." Asked about his disappointment at missing out on Luiz, Arteta continued: "Listen, we had a lot of things still to do in the squad. You see the amount of departures comparing to the players that we have recruited.
"The club has done a phenomenal job and put everything on the table to try to bring the players and the profiles we wanted. We did business pretty early but unfortunately in the last 72 hours, Thomas got injured in the same place he got injured last time and then Mo has a long-term injury and we had to react in the market.
"We tried but at the end of the day, we have to bring the player that we feel is the right player for the team and as well for the club and we didn't manage to do that."
Villa were adamant they would not let Luiz leave, who became an Arsenal target after the club scoured the market with Danilo at Palmeiras one of several players under consideration.
Arteta suggested the club were determined not to overpay or panic late in the window after past mistakes.
"We have our limitations," he said. "Time was a big limitation with what happens in the last 72 hours in our midfield and we tried to act but we didn't find the right player.
"We have learned a lot from that and the players we have to get out, we only have to bring top talent, top quality people through that door and be very disciplined. In those moments, you can make huge mistakes that can cost the club and the team a lot, not in the near future but maybe a little bit ahead and we want to avoid that as much as possible.
"[We need to] believe more in the players we have, look after the. better, give them confidence that they have the ability to do it and that is what we will do."