Ejaria became unsettled in Scotland after some mixed displays.
He was left out of the squad in Europe on Thursday after voicing some problems and returned to Liverpool this weekend.
Davis, despite his age, would offer Rangers some great experience despite dropping back into a deeper midfield role over the last few years.
Former Rangers captain Barry Ferguson has backed the move and thinks Davis would be a great signing.
Rangers news: Steven Davis wants to return to Rangers (Image: GETTY)
Rangers news: Steven Davis has played just twice this season for Southampton (Image: GETTY)
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"The guy just oozes quality," he told PLZ Soccer.
"And you've seen since he left Rangers, he's been in the Premier League a number of years, he's playing at the top level, he's shown his quality.
"I know he's been out of the team this season but he is a player that would make a huge, huge difference to Rangers."
Rangers are hoping to challenge Celtic for the Scottish Premiership title and topped the league last week before losing to Aberdeen and drawing with Dundee to fall back.
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Steven Gerrard reacts to Rangers loss against Rapid Vienna
Celtic are two points above Rangers with a game in hand and both teams play today.
Kilmarnock have gone top again after beating Dundee but have a one point lead having played three more games than Celtic.
Rangers are also in the market for a striker, a creative midfielder and a centre back.
Dominic Solanke is the top striker target but he is also wanted by Premier League clubs while Millwall defender Jake Cooper is on the radar.