Danns is out of contract at Palace and the 28-year-old has been offered a new deal to stay with the Championship club.
However, he is expected to leave Selhurst Park and Gers' Old Firm rivals Celtic, Hull City and Southampton have all been linked with bids.
Bolton Wanderers are the latest club mooted as admirers of Danns, but McCoist is hoping to bring the former Birmingham City player to Ibrox.
"There could be information on Neil Danns in a matter of days," said McCoist in the Daily Record.
"We've been talking to his representatives since he left Crystal Palace and I'm trying to push that one through."
Palace's assistant manager Lennie Lawrence is resigned to Danns moving on, but is confident the club can retain their other key players.
Lawrence told the Croydon Advertiser: "I assume, sadly, that Neil Danns will sign for someone else, but other than that we are not losing anyone else."
Source: Team Talk
Source: Team Talk