The free agent could complete his move to the Hoops today, and is set to be available for the club's Champions League third qualifying round first-leg tie with Astana on Wednesday, The Scottish Sun adds.
However, the former Liverpool man may not be the only defender to arrive at Parkhead before the window closes as the Daily Record claims that Brendan Rodgers is keen on Fenerbahce full-back Hasan Ali Kaldirim.
The 26-year-old is about to begin the final year of his contract, and the Record suggests that he could move on for around £2m.
The Gers have sealed three consecutive wins in the Scottish League Cup without conceding a goal but Halliday clearly felt that the side needed to show more against their lower-league opponents.
"When you come to grounds like this and it is not your familiar surroundings of the 50,000 and the atmosphere you are used to you have to create your own intensity and atmosphere," he told the Evening Times.
"First half, I thought we done that really well but I thought the second half was really disappointing."
Rangers manager Mark Warburton
Mark Warburton has insisted that he was not worried about the fact that they only scored three goals in the game, STV reports.
The group-stages of the League Cup are acting as something of a pre-season for Warburton's men, and he explained that any lack of killer instinct will disappear by the time the season kicks into full swing.
"I would agree that the longer we get into the season the rustiness will ease off and those chances will go in. Against Motherwell last week we could have scored four or five in the second half," he told STV.
"It's fantastic to score on my debut, it was good to get minutes today in my first game and it was really good getting to know my teammates, enjoy the game and to get a goal and a win is fantastic," he told the club's official website.
"I'm watching some Celtic players but I won't tell you who," he told Goal.